diff --git a/-NAyT4oBgHgl3EQfdfcz/content/tmp_files/2301.00302v1.pdf.txt b/-NAyT4oBgHgl3EQfdfcz/content/tmp_files/2301.00302v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6db3ef2eb37e37c608e4016800f350040a92ba9a --- /dev/null +++ b/-NAyT4oBgHgl3EQfdfcz/content/tmp_files/2301.00302v1.pdf.txt @@ -0,0 +1,419 @@ +arXiv:2301.00302v1 [math.CO] 31 Dec 2022 +On Harmonious coloring of hypergraphs +Sebastian Czerwiński +Institute of Mathematics, University of Zielona Góra, Poland +January 3, 2023 +Abstract +A harmonious coloring of a k-uniform hypergraph H is a vertex col- +oring such that no two vertices in the same edge have the same color, +and each k-element subset of colors appears on at most one edge. The +harmonious number h(H) is the least number of colors needed for such a +coloring. +The paper contains a new proof of the upper bounded h(H) = O( +k√ +k!m) +on the harmonious number of k-hypergraphs of maximum degree ∆ with +m edges. We use the local cut lemma of A. Bernshteyn. +1 +Introducion +Let H = (V, E) be a k-uniform hypergraph with the set of vertices V and the +set of edges E. The set of edges is a family of k-elements sets of V , where k ≥ 2. +A rainbow coloring h of a hypergraph H is a map c : V �→ {1, . . . , r} in +which no two vertices in the same edge have the same color. If two vertices are +in the same edge e with the same color, we say that the edge e is bad. +A coloring c is called harmonious if c(e) ̸= c(f) for every pair of distinct +edges e, f ∈ E and c is a rainbow coloring. +We say that distinct edges e and f have the same pattern of colors if c(e\f) = +c(f \ e) and there is no uncolored vertex in the set e \ f. +Let h(H) be the least number of colors needed for a harmonious of H. In +Bosek et al. (2016) proved that +Theorem 1 (Bosek et al. (2016)). For every ε > 0 and every ∆ > 0 there +exist integers k0 and m0 such that every k-uniform hypergraph H with m edges +(where m ≥ m0 and k ≥ k0) and maximum degree ∆ satisfies +h(H) ≤ (1 + ε) +k +k − 1 +k� +∆(k − 1)k!m. +Remark 1. The paper Bosek et al. (2016) contains an upper bound on the +harmonious number +h(H) ≤ +k +k − 1 +k� +∆(k − 1)k!m+1+∆2+(k−1)∆+ +k−1 +� +i=2 +i +i − 1 +i +� +(i − 1)i(k − 1)∆2 +k − i +. +1 + +The proof of this theorem is based on the entropy compression method, see +Grytczuk et al. (2013); Esperet and Parreau (2013). +Because a number r of used colors must satisfy the inequality +�r +k +� +≤ m, we get +lower bound Ω( +k√ +k!m). By these observations, it is conjectured by Bosek et al. +(2016) that +Conjecture 1. For each k, ∆ ≥ 2 there exist a constant c = c(k, ∆) such that +every k-uniform hypergraph H with m edges and maximum degree ∆ satisfies +h(H) ≤ +k√ +k!m + c. +This conjecture was posed by Edwards (1997b) for simple graphs. He prove +there that +h(G) ≤ (1 + o(1)) +√ +2m. +There are many results about the harmonious number of particular classes of +graphs, see Aflaki et al. (2012); Akbari et al. (2012); Edwards (1997a); Edwards and McDiarmid +(1994a); Edwards (1996); Edwards and McDiarmid (1994b); Krasikov and Roditty +(1994) or Aigner et al. (1992); Balister et al. (2002, 2003); Bazgan et al. (1999); +Burris and Schelp (1997). +The paper contains proof of the theorem of Bosek et al., we use a different +method, the local cut lemma of Bernshteyn (2017, 2016). The proof is simpler +and shorter than the original proof of Bosek et al. +2 +A special version of the Local Cut Lemma +Let A be a family of subsets of a powerset Pow(I), where I is a finite set. We +say that it is downwards-closed if for each S ∈ A, implies Pow(S) ⊆ (A). A +subset ∂A of I is called boundary of a downwards-closed family A if +∂A := {i ∈ I : S ∈ A and S ∪ {i} ̸∈ A for some S ⊆ I \ {i}}. +Let τ : T �→ [1; +∞) be a function, then for every X ⊆ I we denote by τ(X) a +number +τ(X) := +� +x∈X +τ(x). +Let B a random event, X ⊆ I and i ∈ I. We introduce two quantities: +σA +τ (B, X) := max +Z⊆I\X Pr(B and Z ∪ X ̸∈ A|Z ∈ A) · τ(X) +and +σA +τ (B, i) := min +i∈X⊆I σA +τ (B, X). +If Pr(Z ∈ A) = 0, then Pr(P|Z ∈ A) = 0 for all events P. +2 + +Theorem 2 (Bernshteyn (2017)). Let I be a finite set. Let Ω be a probabil- +ity space and let A: Ω �→ Pow(Pow(I)) be a random variable such that with +probability 1, A is a nonempty downwards-closed family of subsets of I. For +each i ∈ I, Let B(i) be a finite collection of random events such that whenever +i ∈ ∂A, at least one of the events in B(i) holds. Suppose that there is a function +τ : I �→ [1, +∞) such that for all i ∈ I we have +τ(i) ≥ 1 + +� +B∈B(i) +σA +τ (B, i). +Then Pr(I ∈ A) ≥ 1/τ(I) > 0. +3 +Proof of theorem +We choose a coloring f : V �→ {1, . . ., t} uniformly at random. Let A be a subset +of the power set of V given by +A := {S ⊆ V : c is a harmonious coloring of H(V )}. +It is a nonempty downwards-closed with probability 1 (the empty set is an +element of A) +By a set ∂A, we denote the set of all vertices v such that there is an element +X of A such that the coloring c is not a harmonious coloring of X ∪ {v}. If +the coloring c is not harmonious coloring there is a bad edge or there are two +different edges with the same pattern of colors. So, we define for every v ∈ V a +collection B(v) as union of sets: +B1(v) := {Be : v ∈ e ∈ E(H) and e is not proper colored} +and for every i ∈ {0, 1, . . ., k − 1} +B2 +i (v) := {Be,f : v ∈ e, f ∈ E(H) and c(e) = c(f), |e \ f| = i}. +That is B(v) = B1(v) ∪ �k−1 +i=1 B2 +i (v). +We assume that the event Be happens if and only if the edge e is the bad +edge and the event Be,f happens if and only if edges e and f have the same +pattern of colors. +We also assume that a function τ is a constant function, that is τ(v) = τ ∈ +[1, +∞). This implies that for any subset S of V , we have τ(S) = τ |S|. +Now, we must find an upper bound on +σA +τ (B, v) = +min +X⊆V :v∈X max +Z⊆V \X Pr(B ∧ Z ∪ X ̸∈ A|Z ∈ A)τ(X), +where v ∈ V and B ∈ B(v). +We will be use an estimation σA +τ (B, v) ≤ +maxZ⊆V \X Pr(B|Z ∈ A)τ(X). Now, we consider two cases. +3 + +Case 1: B ∈ B1, i.e. B = Be +We choose as X the set {e}. Because the colors of distinct vertices are indepen- +dent, we get an upper bound σA +τ (Be, v) ≤ Pr(Be)τ k (events Be and ”Z ∈ A” +are independent). The probability Pr(Be) opposite to Pr(Be) full fields +Pr(Be) = 1 − t +t · t − 1 +t +· . . . · t − k + 1 +t +≥ 1 − (1 − k − 1 +t +)k−1. +Through Bernoulli’s inequality, we get +Pr(Be) ≥ 1 − (1 − k − 1 +t +· (k − 1)) = (k − 1)2 +t +. +So, Pr(Be) ≤ k2 +t . +Case 2: B ∈ B2 +i , i.e. B = Be,f and |e \ f| = i +Now, we set X = e \ f. The probability Pr(Be,f) is bonded above by i! +ti . So, we +get +σA +τ (Be,f, v) ≤ Pr(Be,f)τ i ≤ i! +ti τ i. +To end the proof we must find an upper bound on sizes of sets B1(v), B2 +0(v) +and B2 +i (v), where i > 0. Because the degree of a vertex is bounded by above ∆ +and the number of edges is m we get that +|B1(v)| ≤ ∆ and |B2 +0(v)| ≤ ∆m. +The hardest part is an upper bound on B2 +i (v), i > 0. The number of edges +f such that e \ f = i is bounded above by +k∆ +k−i. There are at most k∆ edges +with a nonempty intersection with the edge e and the edge f has exactly k − i +common elements with e. So, we have |B2 +i (v)| ≤ ∆ k∆ +k−i. To apply theorem 2 we +must find τ ∈ [1, +∞) and c ∈ N such that for all v ∈ V below inequality holds +τ ≥ 1 + ∆k2 +t τ k + ∆mk! +tk τ k + +k−1 +� +i=1 +∆ k∆ +k − i +i! +ti τ i. +If we choose τ = +k +k−1 and t = +k +k−1 +k� +∆(k − 1)k!m(1 + ε), it is easy to see that +the inequality holds for sufficiently large hypergraph. +Acknowledgments +References +A. Aflaki, S. Akbari, K. J. Edwards, D. S. Eskandani, M. Jamaali, and H. Ra- +vanbod. On harmonious colouring of trees. Electron. J. Combin., 19(1):Paper +3, 9, 2012. URL https://doi.org/10.37236/9. +4 + +M. Aigner, E. Triesch, and Z. Tuza. +Irregular assignments and vertex- +distinguishing edge-colorings of graphs. In Combinatorics ’90 (Gaeta, 1990), +volume 52 of Ann. Discrete Math., pages 1–9. North-Holland, Amsterdam, +1992. URL https://doi.org/10.1016/S0167-5060(08)70896-3. +S. Akbari, J. Kim, and A. Kostochka. +Harmonious coloring of trees with +large maximum degree. +Discrete Math., 312(10):1633–1637, 2012. +URL +https://doi.org/10.1016/j.disc.2012.02.009. +P. N. Balister, B. Bollobás, and R. H. Schelp. +Vertex distinguishing color- +ings of graphs with ∆(G) = 2. Discrete Math., 252(1-3):17–29, 2002. URL +https://doi.org/10.1016/S0012-365X(01)00287-4. +P. N. Balister, O. M. Riordan, and R. H. Schelp. +Vertex-distinguishing +edge colorings of graphs. +J. Graph Theory, 42(2):95–109, 2003. +URL +https://doi.org/10.1002/jgt.10076. +C. Bazgan, A. Harkat-Benhamdine, H. Li, and M. Woźniak. On the vertex- +distinguishing proper edge-colorings of graphs. J. Combin. Theory Ser. B, 75 +(2):288–301, 1999. 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URL +https://doi.org/10.2298/AADM160411008B. +A. +C. +Burris +and +R. +H. +Schelp. +Vertex-distinguishing +proper +edge-colorings. +J. +Graph +Theory, +26(2):73–82, +1997. +URL +https://doi.org/10.1002/(SICI)1097-0118(199710)26:2<73::AID-JGT2>3.0.CO;2-C. +K. +Edwards. +The +harmonious +chromatic +number +of +bounded +de- +gree +trees. +Combin. +Probab. +Comput., +5(1):15–28, +1996. +URL +https://doi.org/10.1017/S0963548300001802. +K. +Edwards. +The +harmonious +chromatic +number +and +the +achro- +matic number, +volume 241 of London Math. Soc. Lecture Note Ser., +pages +13–47. +Cambridge +Univ. +Press, +Cambridge, +1997a. +URL +https://doi.org/10.1017/CBO9780511662119.003. +K. +Edwards. +The +harmonious +chromatic +number +of +bounded +degree +graphs. +J. +London +Math. +Soc. +(2), +55(3):435–447, +1997b. +URL +https://doi.org/10.1112/S0024610797004857. +5 + +K. +Edwards +and +C. +McDiarmid. +New +upper +bounds +on +harmo- +nious +colorings. +J. +Graph +Theory, +18(3):257–267, +1994a. +URL +https://doi.org/10.1002/jgt.3190180305. +K. +Edwards +and +C. +McDiarmid. +New +upper +bounds +on +harmo- +nious +colorings. +J. +Graph +Theory, +18(3):257–267, +1994b. +URL +https://doi.org/10.1002/jgt.3190180305. +L. Esperet and A. Parreau. +Acyclic edge-coloring using entropy com- +pression. +European +J. +Combin., +34(6):1019–1027, +2013. +URL +https://doi.org/10.1016/j.ejc.2013.02.007. +J. a. Grytczuk, J. Kozik, and P. Micek. +New approach to nonrepetitive +sequences. +Random Structures Algorithms, 42(2):214–225, 2013. +URL +https://doi.org/10.1002/rsa.20411. +I. +Krasikov and +Y. +Roditty. +Bounds +for +the +harmonious +chromatic +number of a graph. +J. Graph Theory, +18(2):205–209, 1994. +URL +https://doi.org/10.1002/jgt.3190180212. +6 + diff --git a/-NAyT4oBgHgl3EQfdfcz/content/tmp_files/load_file.txt b/-NAyT4oBgHgl3EQfdfcz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..98b39417b0aceaeaa2d14615890b93129c8e6fce --- /dev/null +++ b/-NAyT4oBgHgl3EQfdfcz/content/tmp_files/load_file.txt @@ -0,0 +1,341 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf,len=340 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content='00302v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content='CO] 31 Dec 2022 On Harmonious coloring of hypergraphs Sebastian Czerwiński Institute of Mathematics, University of Zielona Góra, Poland January 3, 2023 Abstract A harmonious coloring of a k-uniform hypergraph H is a vertex col- oring such that no two vertices in the same edge have the same color, and each k-element subset of colors appears on at most one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' The harmonious number h(H) is the least number of colors needed for such a coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' The paper contains a new proof of the upper bounded h(H) = O( k√ k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content='m) on the harmonious number of k-hypergraphs of maximum degree ∆ with m edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' We use the local cut lemma of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Bernshteyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' 1 Introducion Let H = (V, E) be a k-uniform hypergraph with the set of vertices V and the set of edges E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' The set of edges is a family of k-elements sets of V , where k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' A rainbow coloring h of a hypergraph H is a map c : V �→ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' , r} in which no two vertices in the same edge have the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' If two vertices are in the same edge e with the same color, we say that the edge e is bad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' A coloring c is called harmonious if c(e) ̸= c(f) for every pair of distinct edges e, f ∈ E and c is a rainbow coloring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' We say that distinct edges e and f have the same pattern of colors if c(e\\f) = c(f \\ e) and there is no uncolored vertex in the set e \\ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Let h(H) be the least number of colors needed for a harmonious of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' In Bosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' (2016) proved that Theorem 1 (Bosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' (2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' For every ε > 0 and every ∆ > 0 there exist integers k0 and m0 such that every k-uniform hypergraph H with m edges (where m ≥ m0 and k ≥ k0) and maximum degree ∆ satisfies h(H) ≤ (1 + ε) k k − 1 k� ∆(k − 1)k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' The paper Bosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' (2016) contains an upper bound on the harmonious number h(H) ≤ k k − 1 k� ∆(k − 1)k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content='m+1+∆2+(k−1)∆+ k−1 � i=2 i i − 1 i � (i − 1)i(k − 1)∆2 k − i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' 1 The proof of this theorem is based on the entropy compression method, see Grytczuk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Esperet and Parreau (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Because a number r of used colors must satisfy the inequality �r k � ≤ m, we get lower bound Ω( k√ k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content='m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' By these observations, it is conjectured by Bosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' (2016) that Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' For each k, ∆ ≥ 2 there exist a constant c = c(k, ∆) such that every k-uniform hypergraph H with m edges and maximum degree ∆ satisfies h(H) ≤ k√ k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content='m + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' This conjecture was posed by Edwards (1997b) for simple graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' He prove there that h(G) ≤ (1 + o(1)) √ 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' There are many results about the harmonious number of particular classes of graphs, see Aflaki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Akbari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Edwards (1997a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Edwards and McDiarmid (1994a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Edwards (1996);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Edwards and McDiarmid (1994b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Krasikov and Roditty (1994) or Aigner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' (1992);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Balister et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' (2002, 2003);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Bazgan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Burris and Schelp (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' The paper contains proof of the theorem of Bosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=', we use a different method, the local cut lemma of Bernshteyn (2017, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' The proof is simpler and shorter than the original proof of Bosek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' 2 A special version of the Local Cut Lemma Let A be a family of subsets of a powerset Pow(I), where I is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' We say that it is downwards-closed if for each S ∈ A, implies Pow(S) ⊆ (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' A subset ∂A of I is called boundary of a downwards-closed family A if ∂A := {i ∈ I : S ∈ A and S ∪ {i} ̸∈ A for some S ⊆ I \\ {i}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Let τ : T �→ [1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' +∞) be a function, then for every X ⊆ I we denote by τ(X) a number τ(X) := � x∈X τ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Let B a random event, X ⊆ I and i ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' We introduce two quantities: σA τ (B, X) := max Z⊆I\\X Pr(B and Z ∪ X ̸∈ A|Z ∈ A) · τ(X) and σA τ (B, i) := min i∈X⊆I σA τ (B, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' If Pr(Z ∈ A) = 0, then Pr(P|Z ∈ A) = 0 for all events P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' 2 Theorem 2 (Bernshteyn (2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Let I be a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Let Ω be a probabil- ity space and let A: Ω �→ Pow(Pow(I)) be a random variable such that with probability 1, A is a nonempty downwards-closed family of subsets of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' For each i ∈ I, Let B(i) be a finite collection of random events such that whenever i ∈ ∂A, at least one of the events in B(i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Suppose that there is a function τ : I �→ [1, +∞) such that for all i ∈ I we have τ(i) ≥ 1 + � B∈B(i) σA τ (B, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Then Pr(I ∈ A) ≥ 1/τ(I) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' 3 Proof of theorem We choose a coloring f : V �→ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=', t} uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Let A be a subset of the power set of V given by A := {S ⊆ V : c is a harmonious coloring of H(V )}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' It is a nonempty downwards-closed with probability 1 (the empty set is an element of A) By a set ∂A, we denote the set of all vertices v such that there is an element X of A such that the coloring c is not a harmonious coloring of X ∪ {v}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' If the coloring c is not harmonious coloring there is a bad edge or there are two different edges with the same pattern of colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' So, we define for every v ∈ V a collection B(v) as union of sets: B1(v) := {Be : v ∈ e ∈ E(H) and e is not proper colored} and for every i ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=', k − 1} B2 i (v) := {Be,f : v ∈ e, f ∈ E(H) and c(e) = c(f), |e \\ f| = i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' That is B(v) = B1(v) ∪ �k−1 i=1 B2 i (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' We assume that the event Be happens if and only if the edge e is the bad edge and the event Be,f happens if and only if edges e and f have the same pattern of colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' We also assume that a function τ is a constant function, that is τ(v) = τ ∈ [1, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' This implies that for any subset S of V , we have τ(S) = τ |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Now, we must find an upper bound on σA τ (B, v) = min X⊆V :v∈X max Z⊆V \\X Pr(B ∧ Z ∪ X ̸∈ A|Z ∈ A)τ(X), where v ∈ V and B ∈ B(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' We will be use an estimation σA τ (B, v) ≤ maxZ⊆V \\X Pr(B|Z ∈ A)τ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Now, we consider two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' 3 Case 1: B ∈ B1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' B = Be We choose as X the set {e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Because the colors of distinct vertices are indepen- dent, we get an upper bound σA τ (Be, v) ≤ Pr(Be)τ k (events Be and ”Z ∈ A” are independent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' The probability Pr(Be) opposite to Pr(Be) full fields Pr(Be) = 1 − t t · t − 1 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' · t − k + 1 t ≥ 1 − (1 − k − 1 t )k−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Through Bernoulli’s inequality, we get Pr(Be) ≥ 1 − (1 − k − 1 t (k − 1)) = (k − 1)2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' So, Pr(Be) ≤ k2 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Case 2: B ∈ B2 i , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' B = Be,f and |e \\ f| = i Now, we set X = e \\ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' The probability Pr(Be,f) is bonded above by i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' ti .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' So, we get σA τ (Be,f, v) ≤ Pr(Be,f)τ i ≤ i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' ti τ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' To end the proof we must find an upper bound on sizes of sets B1(v), B2 0(v) and B2 i (v), where i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Because the degree of a vertex is bounded by above ∆ and the number of edges is m we get that |B1(v)| ≤ ∆ and |B2 0(v)| ≤ ∆m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' The hardest part is an upper bound on B2 i (v), i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' The number of edges f such that e \\ f = i is bounded above by k∆ k−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' There are at most k∆ edges with a nonempty intersection with the edge e and the edge f has exactly k − i common elements with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' So, we have |B2 i (v)| ≤ ∆ k∆ k−i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' To apply theorem 2 we must find τ ∈ [1, +∞) and c ∈ N such that for all v ∈ V below inequality holds τ ≥ 1 + ∆k2 t τ k + ∆mk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' tk τ k + k−1 � i=1 ∆ k∆ k − i i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' ti τ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' If we choose τ = k k−1 and t = k k−1 k� ∆(k − 1)k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content='m(1 + ε), it is easy to see that the inequality holds for sufficiently large hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Acknowledgments References A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Aflaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Akbari, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Edwards, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Eskandani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Jamaali, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Ra- vanbod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' On harmonious colouring of trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=', 19(1):Paper 3, 9, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content='37236/9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' 4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Aigner, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Triesch, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Tuza.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Irregular assignments and vertex- distinguishing edge-colorings of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' In Combinatorics ’90 (Gaeta, 1990), volume 52 of Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' Discrete Math.' metadata={'source': 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+page_content=' Graph Theory, 18(2):205–209, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content='1002/jgt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content='3190180212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} +page_content=' 6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfdfcz/content/2301.00302v1.pdf'} diff --git a/-dAzT4oBgHgl3EQf_P7B/content/tmp_files/2301.01946v1.pdf.txt b/-dAzT4oBgHgl3EQf_P7B/content/tmp_files/2301.01946v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1fd43eaa247510ffd4557ce153cc710752373286 --- /dev/null +++ b/-dAzT4oBgHgl3EQf_P7B/content/tmp_files/2301.01946v1.pdf.txt @@ -0,0 +1,2040 @@ +EPR-Net: Constructing non-equilibrium potential landscape via a variational force +projection formulation +Yue Zhao,1 Wei Zhang,2, ∗ and Tiejun Li1, 3, 4, † +1Center for Data Science, Peking University, Beijing 100871, China +2Zuse Institute Berlin, D-14195 Berlin, Germany +3LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China +4Center for Machine Learning Research, Peking University, Beijing 100871, China +(Dated: January 6, 2023) +We present a novel yet simple deep learning approach, dubbed EPR-Net, for constructing the +potential landscape of high-dimensional non-equilibrium steady state (NESS) systems. The key idea +of our approach is to utilize the fact that the negative potential gradient is the orthogonal projection +of the driving force in a weighted Hilbert space with respect to the steady-state distribution. The +constructed loss function also coincides with the entropy production rate (EPR) formula in NESS +theory. This approach can be extended to dealing with dimensionality reduction and state-dependent +diffusion coefficients in a unified fashion. The robustness and effectiveness of the proposed approach +are demonstrated by numerical studies of several high-dimensional biophysical models with multi- +stability, limit cycle, or strange attractor with non-vanishing noise. +Since Waddington’s famous landscape metaphor on the +development of cells in the 1950s [1], the construction of +potential landscape for non-equilibrium biochemical reac- +tion systems has been recognized as an important prob- +lem in theoretical biology, as it provides insightful pic- +tures for understanding complex dynamical mechanisms +of biological processes. This problem has attracted con- +siderable attention in recent decades in both biophysics +and applied mathematics community. Until now, several +approaches have been proposed to realize Waddington’s +landscape metaphor in a rational way, see [2–10] and +the references therein for details and [11–14] for reviews. +Broadly speaking, these proposals can be classified into +two types: (T1) the construction of potential landscape +in the finite noise regime [3–5] and (T2) the construction +of the quasi-potential in the zero noise limit [2, 6–9]. +For low-dimensional systems (i.e., dimension less than +4), the potential landscape can be numerically computed +either by solving a Fokker-Planck equation (FPE) using +grid-based methods until the steady solution is reached +approximately as in (T1) type proposals [3, 5], or by solv- +ing a Hamilton-Jacobi-Bellman (HJB) equation using, for +instance, the ordered upwind method [15] or minimum +action type method [8] as in (T2) type proposals. How- +ever, these approaches suffer from the curse of dimen- +sionality when applied to high-dimensional systems. Al- +though methods based on mean field approximations are +able to provide a semi-quantitative description of the en- +ergy landscape for typical systems [4, 16], direct and gen- +eral approaches are still favored in applications. In this +aspect, pioneering work has been done recently, which +allows direct construction of high-dimensional potential +landscape using deep neural networks (DNN), based on +either the steady viscous HJB equation satisfied by the +∗ wei.zhang@fu-berlin.de +† tieli@pku.edu.cn +landscape function in (T1) case [17, 18], or the point- +wise orthogonal decomposition of the force field in (T2) +case [19]. These works have brought significant advances +in the methodological developments in both cases. How- +ever, these approaches, which are based on solving HJB +equations alone, may encounter numerical difficulties due +to the non-uniqueness of the weak solution to the non- +viscous HJB equation in (T2) case [20], and challenges +in solving the steady HJB equation with a small noise in +(T1) case. +Setup. +In this letter, we present a simple yet ef- +fective DNN approach, EPR-Net, for constructing the +potential landscape of high-dimensional non-equilibrium +steady state (NESS) systems in (T1) type. Our key ob- +servation is that the negative potential gradient is the or- +thogonal projection of the driving force under a weighted +inner product with respect to the steady-state distribu- +tion. To be specific, let us consider the stochastic differ- +ential equations (SDEs) +dx(t) +dt += F (x(t)) + +√ +2D ˙w, +x(0) = x0, +(1) +where x0 ∈ Rd, F : Rd → Rd is a smooth function, +˙w = ( ˙w1, . . . , ˙wd)⊤ is the d-dimensional temporal Gaus- +sian white noise with E ˙wi(t) = 0 and E[ ˙wi(t) ˙wj(s)] = +δijδ(t − s) for i, j = 1, . . . , d, s, t > 0 and D > 0 is the +noise strength, which is often related to the system’s tem- +perature T by D = kBT, where kB is the Boltzmann con- +stant. We assume that (1) is ergodic and denote by pss(x) +its steady-state probability density function (PDF). +We follow the (T1) type proposal in [3] to derive the po- +tential landscape of (1) in the case of D > 0. That is, we +define the potential U = −D ln pss and the steady proba- +bility flux Jss = pssF −D∇pss in the domain Ω, which we +assume for simplicity is either Rd or a d-dimensional hy- +perrectangle. The steady-state PDF pss(x) satisfies the +Fokker-Planck equation (FPE) +∇ · (pssF ) − D∆pss = 0, +for x ∈ Ω, +(2) +arXiv:2301.01946v1 [physics.bio-ph] 5 Jan 2023 + +2 +and we assume the asymptotic boundary condition (BC) +pss(x) → 0 as |x| → ∞ when Ω = Rd, or the re- +flecting boundary condition Jss · n = 0 when Ω ⊂ Rd +is a d-dimensional hyperrectangle, where n is the unit +outer normal. +In both cases, we have pss(x) ≥ 0 and +� +Ω pss(x) dx = 1. +Learning approach. Aiming at an effective approach +for high-dimensional applications, we employ DNNs to +approximate U(x), and the key idea in this letter is to +learn U by training DNN with the following loss function +LEPR(V ) = +� +Ω +|F (x) + ∇V (x; θ)|2 dπ(x), +(3) +where V := V (x; θ) is a neural network function with +parameters θ [21], and dπ(x) = pss(x) dx. +To justify +(3), we note that U satisfies the important orthogonality +relation: for any suitable function W : Rd → R, +� +Ω +� +F (x) + ∇U(x) +� +· ∇W(x) dπ(x) = 0. +(4) +Therefore, U(x) is the unique minimizer (up to a con- +stant) of the loss LEPR and, moreover, the negative po- +tential gradient −∇U is in fact the projection of the force +field F in the π-weighted Hilbert space. See Sec. A and B +in the Supplemental Material (SM) for derivations in de- +tail. +The minimum loss LEPR(U) has a clear physical inter- +pretation. Indeed, we have (see SM Sec. B) +LEPR(U) = +� +Ω +|Jss|2 1 +pss +dx = ess +p , +(5) +where ess +p denotes the steady entropy production rate +(EPR) of the NESS system (1) [3, 22, 23]. Therefore, +minimizing (3) is equivalent to approximating the steady +EPR. This explains the name EPR-Net of our approach. +To utilize (3) in numerical computations, we replace +the spatial integral in (3) with respect to the unknown π +by its empirical average using data sampled from (1): +�LEPR(θ) = 1 +N +N +� +i=1 +��F (xi) + ∇V (xi; θ) +��2, +(6) +where (xi)1≤i≤N could be either the final states (at time +T) of N trajectories starting from different initializations +or equally spaced time series along a single long trajec- +tory up to time T, where T ≫ 1. +In both cases, the +ergodicity of SDE (1) guarantees that (6) is a good ap- +proximation of (3) as long as T is large [24]. We adopt +the former approach in the numerical experiments in this +work, where the gradients of both V (with respect to x) +and the loss itself (with respect to θ) in (6) are calculated +by auto-differentiation through PyTorch [25]. The stabil- +ity analysis of this approximation is presented in detail +in SM Sec. C. +We apply our method to a toy model first in order to +check its applicability and accuracy. We take +F (x) = −(I + A) · ∇U0(x), +(7) +where A ∈ Rd×d is a constant skew-symmetric matrix, +i.e., A⊤ = −A, and U0 is some known function. With this +choice of F , one can check that the true potential land- +scape is U(x) = U0(x). In particular, the system is re- +versible when A = 0. Based on the proposed method, we +construct a double-well model with known potential U0 +for verification. We take D = 0.1. As shown in Fig. 1(A), +the learned potential agrees well with the simulated sam- +ples. +Also, the decomposition of the force field shows +that the negative gradient part −∇V (x; θ) around the +wells points towards the attractor and is nearly orthog- +onal to the non-gradient part. The overall non-gradient +field shows a counter-clockwise rotation. +The relative +root mean square error (rRMSE) of the potential V (x; θ) +learned by EPR loss is 0.0987 (averaged over 3 runs), +which supports the effectiveness of our approach. +See +SM Sec. F F.1 for details of the problem setting. +The correct interpretation of the computational results +based on the EPR loss (3) is that the accuracy of V (x) +is guaranteed only when π(x) is evidently above zero +for any specific x. +In the “visible” domain of π (i.e., +the places where there are sample points of {xi}), the +trained potential V gives reliable approximation; while +in the weakly visible or invisible domain, especially in +local transition regions between meta-stable states and +boundaries of the visible domain, we must resort to the +original FPE (2) which holds pointwise in space. +Learning strategy for small D. Substituting the rela- +tion pss(x) = exp(−U(x)/D) into (2), we get the viscous +HJB equation +NHJB(U) := F · ∇U + |∇U|2 − D∆U − D∇ · F = 0 (8) +with the asymptotic BC U → ∞ as |x| → ∞ in the case +of Ω = Rd, or the reflecting BC (F + ∇U) · n = 0 on ∂Ω +when Ω is a d-dimensional hyperrectangle, respectively. +As in the framework of physics-informed neural networks +(PINNs) [26], (8) motivates the HJB loss +LHJB(V ) = +� +Ω +��NHJB(V (x; θ)) +��2 dµ(x), +(9) +where µ is any desirable distribution. +By choosing µ +properly, this loss allows the use of sample data that +better cover the domain Ω and, when combined with the +loss in (3), leads to significant improvement of the train- +ing results in our numerical experiments when D is small. +Specifically, for small D, we propose the enhanced loss in +training which has the form +�Lenh(θ) = �LEPR(θ) + λ �LHJB(θ), +(10) +where +�LEPR(θ) +is +defined +in +(6), +�LHJB(θ) += +1 +N ′ +�N ′ +i=1 |NHJB(V (x′ +i; θ))|2 is an approximation of (9) us- +ing an independent data set (x′ +i)1≤i≤N ′ obtained by sam- +pling the trajectories of (1) with a larger D′ > D, and +λ > 0 is a weight parameter balancing the contribution +of the two terms in (10). Note that the proposed strategy +is both general and easily adaptable. For instance, one + +3 +FIG. 1. Filled contour plots of the learned potential V (x; θ) for (A) toy model learned by EPR loss (3) with D = 0.1, and +(B)-(C) a biochemical oscillation network model [3] and a tri-stable cell development model [5] learned by enhanced loss (10). +The force field F (x) is decomposed into the gradient part −∇V (x; θ) (white arrows) and the non-gradient part (gray arrows). +The length of an arrow denotes the scale of the vector. The solid dots are samples from the simulated invariant distribution. +can alternatively use data (x′ +i)1≤i≤N ′ that contains more +samples in the transition region, or employ a modification +of the loss (9) in (10) [17]. +We apply our enhanced loss (10) to construct the land- +scape for a 2D biological system with a limit cycle [3] +and a 2D multistable system [5]. The potential V (x; θ) +learned by the enhanced loss (10), the force decomposi- +tion, and sample points from the simulated invariant dis- +tribution are shown in Fig. 1(B) and (C). As in the toy +model case, the gradient part (white arrows) points di- +rectly towards the attractors, while the non-gradient part +(gray arrows) shows a counter-clockwise rotation for the +limit cycle, and a splitting-and-back flow from the mid- +dle attractor to the other two attractors for the tri-stable +dynamical model. To further verify the accuracy of the +method, we numerically solve the FPE (2) as reference +solutions by a fine grid discretization. Comparisons be- +tween the proposed method and the method based on +the naive HJB loss on these two problems are demon- +strated in SM. Averaged over 3 runs, the rRMSE of the +potential V learned by our enhanced loss is 0.0524 and +0.0402, respectively, which shows an evident advantage +over the naive HJB loss. See SM Sec. F for details of the +comparisons. +Dimensionality reduction. +When applying the ap- +proach above to high-dimensional problems, dimensional- +ity reduction is necessary in order to visualize the results +and gain physical insights. A straightforward approach is +to first learn the high-dimensional potential U and then +find its low-dimensional representation, i.e., the reduced +potential or the free energy function, using dimension- +ality reduction techniques (see SM Sec. D D.1). In the +following, we present an alternative approach that allows +to directly learn the low-dimensional reduced potential. +For simplicity, we consider the linear case and, with a +slight abuse of notation, denote by x = (y, z)⊤, where +z = (xi, xj) ∈ R2 contains the coordinates of two vari- +ables of interest, and y ∈ Rd−2 corresponds to the +other d − 2 variables. +The domain Ω (either Rd or +a d-dimensional hyperrectangle) has the decomposition +Ω = Σ × �Ω, where Σ ⊆ Rd−2 and �Ω ⊆ R2 are the do- +mains of y and z, respectively. As can be seen in the +numerical examples, this setting is applicable to many +interesting biochemical systems. Extensions to nonlinear +low-dimensional reduced variables with general domains +are possible, e.g., by applying the approach developed +in [27]. In the current setting, the reduced potential is +�U(z) = −D ln �pss(z) = −D ln +� +Σ +pss(y, z) dy, +(11) +and one can show that �U minimizes the following loss +function: +LP-EPR(�V ) = +� +Ω +��Fz(y, z)+∇z �V (z; θ) +��2 dπ(y, z), (12) +where Fz(y, z) ∈ R2 is the z-component of the force +field F = (Fy, Fz)⊤. Similar to (6), the empirical form +of (12) can be used in learning the reduced potential �U. +Moreover, one can derive an enhanced loss as in (10) that +could be used for systems with small D. To this end, we +note that �U satisfies the projected HJB equation +NP-HJB(�U) := �F · ∇z �U + |∇z �U|2 +− D∆z �U − D∇z · �F = 0 , +(13) +with asymptotic BC �U +→ ∞ as |z| → ∞, or the +reflecting BC ( �F + ∇z �U) · �n += +0 on ∂�Ω, where +�F (z) +:= +� +Σ Fz(y, z)dπ(y|z) is the projected force +defined using the conditional distribution dπ(y|z) = +pss(y, z)/�pss(z) dy, and �n denotes the unit outer normal +on ∂�Ω. Based on (13), we can formulate the projected +HJB loss +LP-HJB(�V ) = +� +�Ω +��NP-HJB(�V (z; θ)) +��2 dµ(z), +(14) + +A +B +3.0 +2.00 +2.00 +0.20 +8 +2.5 +7 +2.5 +1.60 +1.60 +0.16 +6 +2.0 +2.0 +5 +0.12 +1.20 +1.20 +1.5 +>1.5 +4 +0.04 +0.80 +-0.80 +1.0 +3 +1.0- +2 +-0.40 0.5 +0.40 +0.04 +0.5 +0.00 +0.0 +0.00 +-0.00 0.0 +0 +0.5 +2.5 +6 +0.5 +2.0 +1.5 +2.0 +2 +8 +1.0 +1.5 +2.5 +0.0 +1.0 +3.0 +0 +4 +0.0 ++ +X +X4 +where µ is any suitable distribution over �Ω, and �F in (13) +is learned beforehand by training a DNN with the loss +LP-For( � +G) = +� +Ω +��Fz(y, z) − � +G(z; θ) +��2 dπ(y, z). +(15) +The overall enhanced loss used in numerical computa- +tions comprises two terms, which are empirical estimates +of (12) and (14) based on two different sets of sample +data. See SM Sec. D for derivation details. +We then apply our dimensionality reduction approach +to construct the landscape for an 8D cell cycle model con- +taining both a limit cycle and a stable equilibrium point +for the chosen parameters, and take CycB and Cdc20 as +the reduced variables following [4]. As shown in Fig. 2, we +can find that the depth of the reduced potential and force +strength agree well with the density of projected samples. +Moreover, we can also get some important insights from +Fig. 2 on the projection of the high-dimensional dynam- +ics with a limit cycle to two dimensions. One particular +feature is that the limit cycle induced by the projected +force � +G (outer red circle) has minor differences with the +limit cycle directly projected from high dimensions (yel- +low circle), and the difference is slight or moderate de- +pending on whether the density of samples is high or +low. This is natural in the reduction since the distribu- +tion π(y|z) in the projection is not of Dirac type when +D > 0, and this difference will disappear as D → 0. +Another feature is that we unexpectedly get an addi- +tional stable limit cycle (inner red circle) and a stable +point (red dot in the center) emerging inside the limit +cycle. +Though virtual in high dimensions and biologi- +cally irrelevant, the existence of such two limit sets is +reminiscent of the Poincar´e-Bendixson theorem in pla- +nar dynamics theory [28, Chapter 10.6], which depicts +a common phenomenon when performing dimensionality +reduction with limit cycles to 2D plane. The emergence +of these two limit sets, though being not a general sit- +uation, is specific in the considered model due to the +relatively flat landscape of the potential in the centering +region. In addition, close to the saddle point (0.13, 0.55) +of �V (green star), there is a barrier domain along the +limit cycle direction, while a local well domain along the +Cdc20 direction, which characterizes the region that bi- +ological cycle paths mainly go through. +Last but not +the least, a zoom-in view of the local well domain out- +side of the limit cycle shows its detailed spiral structure +(Fig. 2C), which has not been revealed before by mak- +ing a Gaussian approximation. Some other applications +of our approach to Ferrell’s three-ODE model [29], 52D +stem cell network model [16] and 3D Lorenz model are +demonstrated in SM Sec. G and H. +Extension to variable diffusion coefficient case. +The +EPR-Net formulation can be extended to the case of +state-dependent diffusion coefficients without any diffi- +culty. Consider the Ito SDEs +dx(t) +dt += F (x(t)) + +√ +2Dσ(x(t)) ˙w, +x(0) = x0, +(16) +FIG. 2. Dimensionality reduction of an 8D cell cycle model +with two reduced variables. (A) Reduced potential landscape +�V with projected contour lines. (B) Projected sample points, +streamlines of the projected force field � +G and the filled con- +tour plot of �V . The red circles and dots are stable limit sets +of the projected force field. The yellow circle is the projection +of the original high-dimensional limit cycle. (C) The detailed +spiral structure of the streamlines of � +G around the stable +point by zooming in the square domain in (B). +with diffusion matrix σ(x) ∈ Rd×m and +˙w is an m- +dimensional temporal Gaussian white noise. We assume +that m ≥ d and the matrix a(x) := (σσ⊤)(x) satisfies +u⊤a(x)u ≥ c0|u|2 for all x, u ∈ Rd, where c0 > 0 is a +positive constant. Using a similar derivation as before, +we can again show that the high-dimensional landscape +function U of (16) minimizes the EPR loss +LV-EPR(V ) = +� +Ω +|F v(x) + a(x)∇V (x)|2 +a−1(x) dπ(x), +(17) +where F v(x) = F (x) − D∇ · a(x) and |u|2 +a−1(x) := +u⊤a−1(x)u for u ∈ Rd. We provide derivation details +of (17) in SM Sec. E. However, we will not pursue a nu- +merical study of (16)–(17) in this paper. +Discussions and Conclusion. Below we make some fi- +nal remarks. First, concerning the use of the steady-state +distribution π(x) in (3) and its approximation by a long +time series of the SDE (1) in EPR-Net, we emphasize that +it is the sampling approximation of π that naturally cap- +tures the important parts of the potential function, and +the landscape beyond the sampled regions is not that +essential in practice. +Second, as is exemplified in SM +Sec. F F.4, we found that a direct application of density +estimation methods (DEM), e.g., normalizing flows [30], +to the sampled time series data does not give potential + +A +B +1.0 +0.08 +0.08 +0.07 +0.06 +0.04 +0.06 +0.02 +0.05 +0.00 +-0.02 +0.8 +0.04 +-0.04 +0.03 +1.0 +0.8 +0.02 +0.60 +0.01 +0.4 +0.2 +0.0 +0.1 +0.2 +0.6 +0.3 +0.4 +0.5 +CycB +Cdc20 +C +3.0 +0.16 +0.4 +2.5 +0.14 +2.0 +0.12 +1.5 +0.10 +0.2 +1.0 +0.08 +0.5 +0.06 +0.18 +0.20 +0.22 +0.24 +0.26 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +CycB5 +landscape with satisfactory accuracy. We speculate that +such deficiency of DEM is due to its over-generality and +the fact that it does not take advantage of the force field +information explicitly compared to (3). +Overall, we have presented the EPR-Net, a simple +yet effective DNN approach, for constructing the non- +equilibrium potential landscape of NESS systems. This +approach is both elegant and robust due to its variational +structure and its flexibility to be combined with other +types of loss functions. Further extension of dimensional- +ity reduction to nonlinear reduced variables and numeri- +cal investigations in the case of state-dependents diffusion +coefficients will be explored in future work. +Acknowledgement. +We thank Professors Chunhe Li, +Xiaoliang Wan and Dr. Yufei Ma for helpful discus- +sions. TL and YZ acknowledge the support from NSFC +and MSTC under Grant No.s 11825102, 12288101 and +2021YFA1003300. +WZ is supported by the DFG un- +der Germany’s Excellence Strategy-MATH+: The Berlin +Mathematics Research Centre (EXC-2046/1)-project ID: +390685689. +The numerical computations of this work +were conducted on the High-performance Computing +Platform of Peking University. + +6 +Supplemental Material for: +EPR-Net: Constructing non-equilibrium potential landscape via +a variational force projection formulation +CONTENTS +Part 1: Theory +6 +A. Validation of the EPR loss +6 +B. EPR loss and entropy production rate +7 +C. Stability of the EPR minimizer +7 +D. Dimensionality reduction +8 +D.1. Gradient projection loss +8 +D.2. Projected EPR loss +8 +D.3. Force projection loss +9 +D.4. HJB equation for the reduced potential +9 +E. State-dependent diffusion coefficients +9 +Part 2: Computation +10 +F. 2D models and comparisons +10 +F.1. Toy model and enhanced EPR +10 +F.2. 2D limit cycle model +11 +F.3. 2D multi-stable model +12 +F.4. Numerical comparisons +12 +G. 3D models +13 +G.1. 3D Lorenz system +14 +G.2. Ferrell’s three-ODE model +14 +H. High dimensional models +15 +H.1. 8D complex system +15 +H.2. 52D multi-stable system +16 +References +17 +In this supplemental material (SM), we will present +further theoretical derivations and computational details +of the contents in the main text (MT). This SM consists +of two parts: Theory and computation. +PART 1: THEORY +We will first provide details of theoretical derivations +omitted in the MT. +A. +VALIDATION OF THE EPR LOSS +In this section, we show that, up to an additive con- +stant, the potential function U(x) := −D ln pss(x) is the +unique minimizer of the EPR loss (3) defined in the MT. +First, we show that the orthogonality relation +� +Ω +(F + ∇U) · ∇W dπ = 0 +(18) +holds for any suitable function W(x) : Rd → R under +both choices of the boundary conditions (BC) considered +in the MT, where dπ(x) := pss(x)dx. To see this, we +note that +� +Ω +(F + ∇U) · ∇W dπ += +� +Ω +(F pss − D∇pss) · ∇W dx += +� +∂Ω +W(F pss − D∇pss) · n dx +− +� +Ω +W∇ · (F pss − D∇pss) dx +:=P1 − P2 +where we have used integration by parts and the relation +pss(x) = exp(−U(x)/D). +The term P1 is zero due to +the fact that pss(x) tends to 0 exponentially as |x| → ∞ +when Ω = Rd, and the reflecting BC Jss · n = 0 which +holds on ∂Ω when Ω is bounded. The term P2 is zero +due to the steady state Fokker-Planck equation (FPE) +satisfied by pss. +Now consider the EPR loss, we have +LEPR(V ) = +� +Ω +|F + ∇V |2 dπ += +� +Ω +|F + ∇U + ∇V − ∇U|2 dπ += +� +Ω +� +|F + ∇U|2 + |∇V − ∇U|2� +dπ ++ 2 +� +Ω +(F + ∇U) · ∇(V − U) dπ += +� +Ω +|F + ∇U|2 + |∇V − ∇U|2 dπ, +where we have used the orthogonality relation (18) to +arrive at the last equality, from which we conclude that + +7 +U(x) is the unique minimizer of the EPR loss up to an +additive constant. +In fact, define the π-weighted inner product for any +square integrable functions f, g on Ω: +(f, g)π := +� +Ω +f(x)g(x) dπ(x) +(19) +and the corresponding L2 +π-norm ∥·∥π by ∥f∥2 +π := (f, f)π, +we get a Hilbert space L2 +π (see, e.g., [31, Chapter II.1]). +Choosing W = U in (18), we observe that the minimiza- +tion of EPR loss finds the orthogonal projection of F +under the π-weighted inner product, i.e., +F (x) = −∇U(x) + l(x), such that (∇U, l)π = 0. (20) +However, we remark that this orthogonality holds only +in the L2 +π-inner product sense instead of the pointwise +sense. Furthermore, the two orthogonality relations (18) +and (20) can be understood as follows. Using (20), the +relation (18) is equivalent to +� +Ω l · ∇Wdπ = 0 for any +W. Integration by parts gives ∇ · (l e−U/D) = 0, which +is equivalent to ∇U · l + D∇ · l = 0. When D → 0, we +recover the pointwise orthogonality, which is adopted in +computing quasi-potentials in [19]. +B. +EPR LOSS AND ENTROPY PRODUCTION +RATE +In this section, we show that the minimum EPR loss +coincides with the steady entropy production rate in non- +equilibrium steady state (NESS) theory. +Following [22, 23], we have the important identity con- +cerning the entropy production for the SDE (1) defined +in the MT: +DdS(t) +dt += ep(t) − hd(t), +(21) +where S(t) := − +� +Ω p(x, t) ln p(x, t) dx is the entropy of +the probability density function p(x, t) at time t, ep is +the entropy production rate (EPR) +ep(t) = +� +Ω +|F (x) − D∇ ln p(x, t)|2 p(x, t) dx, +(22) +and hd is the heat dissipation rate +hd(t) = +� +Ω +F (x) · J(x, t) dx, +(23) +with the probability flux J(x, t) +:= +p(x, t)(F (x) − +D∇ ln p(x, t)) at time t. When D = kBT, the above for- +mulas have clear physical meaning in statistical physics. +At the steady state, we get the steady EPR +ess +p = +� +Ω +|F − D∇ ln pss|2 pss dx += +� +Ω +|F + ∇U|2 pss dx += +� +Ω +|Jss|2 1 +pss +dx = LEPR(U), +where Jss(x) = pss(x)(F (x)+∇U(x)) is the steady prob- +ability flux. +This shows the relation between the pro- +posed EPR loss function and the entropy production rate +in the NESS theory. +C. +STABILITY OF THE EPR MINIMIZER +In this section, we formally show that small perturba- +tions of the invariant distribution π will not introduce +a disastrous change to the minimizer of the correspond- +ing EPR loss. We only consider the bounded domain, +i.e., Ω is a hyperrectangle. The argument for unbounded +domains is similar. +Suppose dπ(x) = p(x)dx, dµ(x) = q(x)dx, and the +functions U(x) and ¯U(x) are the unique minimizers (up +to a constant) of the following two EPR losses +U = arg min +V +� +Ω +|F + ∇V |2 dπ, +¯U = arg min +V +� +Ω +|F + ∇V |2 dµ, +respectively. +It is not difficult to find that the Euler- +Lagrange equations of U, ¯U are given by the following +partial differential equation (PDE) with suitable BCs: +∇ · ((F + ∇U)p) = 0 in Ω, (F + ∇U) · n = 0 on ∂Ω, +∇ · ((F + ∇ ¯U)q) = 0 in Ω, (F + ∇ ¯U) · n = 0 on ∂Ω. +The PDEs above defined inside the domain Ω can be +converted to +∆Up + ∇U · ∇p = −∇ · (pF ), +∆ ¯Uq + ∇ ¯U · ∇q = −∇ · (qF ). +Define U0(x) = −D ln p(x) and ¯U0(x) = −D ln q(x). We +then obtain +−∇U · ∇U0 + D∆U = F · ∇U0 − D∇ · F , +(24) +−∇ ¯U · ∇ ¯U0 + D∆ ¯U = F · ∇ ¯U0 − D∇ · F . +(25) +Assuming that δU0 := U0 − ¯U0 = O(ε), where 0 < ϵ ≪ 1 +denotes a small constant, we have the PDE for U − ¯U by +subtracting (25) from (24): +−∇(U− ¯U) · ∇U0 + D∆(U − ¯U) += F · ∇(δU0) + ∇ ¯U · ∇(δU0) +with BC ∇(U − ¯U) · n = 0. Since U0, ¯U, F ∼ O(1), we +can obtain that +U(x) − ¯U(x) = O(ε) +by the regularity theory of elliptic PDE [32, Section 6.3] +when D ∼ O(1), or by the matched asymptotic expan- +sion when D ≪ 1 [33, Chapter 2]. In fact, the closeness +between U(x) and ¯U(x) can be ensured as long as U0 and +¯U0 are close enough in the region where p(x) and q(x) are +bounded away from zero by the method of characteristics +analysis [32, Section 2.1] and matched asymptotics. + +8 +D. +DIMENSIONALITY REDUCTION +In this section, we study dimensionality reduction for +high-dimensional problems in order to learn the projected +potential. +Denote by x = (y, z)⊤ ∈ Ω. As in the MT, we assume +the domain +Ω = �Ω × Σ, +where �Ω ⊆ R2 and Σ ⊆ Rd−2 are the domain of y and z, +respectively. The reduced potential �U(z) is defined as +�U(z) = −D ln �pss(z) = −D ln +� +Σ +pss(y, z) dy. +(26) +One natural approach for constructing �U(z) is directly +integrating pss(y, z) based on the learned U(y, z) with +the EPR loss, i.e., +�U(z) = −D ln +� +Σ +exp(−U(y, z)/D) dy. +(27) +However, performing this integration is not a straightfor- +ward numerical task (see, e.g., [34, Chapter 7]). +D.1. +Gradient projection loss +In this subsection, we study a simple approach to +approximate �U(z) based on sample points, which ap- +proximately obey the invariant distribution π(x), and +the learned high dimensional potential function U(x) by +EPR loss. This approach is not investigated numerically +in this work, but it will be useful for the derivations in +the next subsection. The idea is to utilize the gradient +projection (GP) loss on the z components of ∇U: +LGP(�V ) = +� +Ω +��∇zU(y, z) − ∇z �V (z) +��2 dπ(y, z). +(28) +To justify (28), we note that +LGP(�V ) = +� +Ω +��∇zU − ∇z �V +��2 dπ(x) += +� +Ω +��∇zU − ∇z �U + ∇z �U − ∇z �V +��2 dπ(x) += +� +Ω +���∇zU − ∇z �U +��2 + +��∇z �U − ∇z �V +��2� +dπ(x) ++ 2 +� +Ω +(∇zU − ∇z �U) · ∇z(�U − �V ) dπ(x) +=:P1 + P2, +where P1 and P2 denote the terms in the third and the +fourth line above, respectively. The term P2 = 0 since +� +Ω +∇zU · ∇z(�U − �V ) dπ(x) += +� +�Ω +�� +Σ +∇zUe− U +D dy +� +· ∇z(�U − �V ) dz += − D +� +�Ω +∇z +�� +Σ +e− U +D dy +� +· ∇z(�U − �V ) dz += − D +� +�Ω +∇z�pss · ∇z(�U − �V ) dz += +� +�Ω +∇z �U · ∇z(�U − �V ) �pss dz +and +� +Ω +∇z �U · ∇z(�U − �V ) dπ(x) += +� +�Ω +∇z �U · ∇z(�U − �V ) �pss dz, +which cancel with each other in P2. +Therefore, the minimization of GP loss is equivalent to +minimizing +� +�Ω +��∇z �U − ∇z �V +��2 �pss dz, +which clearly implies that �U(z) is the unique minimizer +(up to a constant) of the proposed GP loss. +D.2. +Projected EPR loss +In this subsection, we study the projected EPR (P- +EPR) loss, which has the form +LP-EPR(�V ) = +� +Ω +��Fz(y, z) + ∇z �V (z) +��2 dπ(y, z), +(29) +where Fz(y, z) ∈ R2 is the z-component of the force field +F = (Fy, Fz)⊤. +Define +�LP-EPR(�V ) = +� +Ω +��F (y, z) + ∇�V (z) +��2 dπ(y, z), +(30) +where ∇ is the full gradient with respect to x. To justify +(29), we first note the following equivalence +min LP-EPR(�V ) +⇐⇒ +min �LP-EPR(�V ), +(31) +since ∇y �V (z) = 0 and the y-components of F +∇�V only +introduce an irrelevant constant in (30). Furthermore, we +have +�LP-EPR(�V ) = +� +Ω +��F + ∇�V +��2 dπ(x) += +� +Ω +��F + ∇U + ∇�V − ∇U +��2 dπ(x) += +� +Ω +��F + ∇U +��2 + +��∇�V − ∇U +��2 dπ(x), + +9 +where the last equality is due to the orthogonality rela- +tion (18). Using a similar argument for deriving (31), the +equivalence (31) itself, as well as the GP loss in (28), we +get +min LP-EPR(�V ) +⇐⇒ +min LGP(�V ). +(32) +Since �U minimizes the GP loss as is shown in the previous +subsection, we conclude that �U minimizes the loss in (29). +D.3. +Force projection loss +In this subsection, we study the force projection (P- +For) loss for approximating the projection of Fz onto the +z-space. +Denote by +�F (z) := +� +Σ +Fz(y, z) dπ(y|z) +(33) +the projected force defined using the conditional distri- +bution +dπ(y|z) = pss(y, z)/�pss(z) dy. +(34) +We can learn �F (z) via the following force projection loss +LP-For( � +G) = +� +Ω +��Fz(y, z) − � +G(z) +��2 dπ(y, z). +(35) +To justify (35), we note that +� +Ω +��Fz(y, z) − � +G(z) +��2 dπ(y, z) += +� +Ω +� +|Fz(y, z)|2 + | � +G(z)|2� +dπ(y, z) +− 2 +� +Ω +Fz(y, z) · � +G(z) dπ(y, z) +=:P1 − 2P2. +The term P2 can be simplified as +P2 = +� +�Ω +�� +Σ +Fz(y, z)dπ(y|z) +� +· � +G(z) �pss(z) dz += +� +�Ω +�F (z) · � +G(z) �pss(z) dz. +Therefore, we have the equivalence +min LP-For( � +G) +⇐⇒ +min �LP-For( � +G), +(36) +where +�LP-For( � +G) := +� +�Ω +�� �F (z) − � +G(z) +��2�pss(z) dz. +From the analysis above we can conclude that �F(z) min- +imizes the loss in (35). +D.4. +HJB equation for the reduced potential +In this subsection, we show that the reduced potential +�U satisfies the projected HJB equation +�F · ∇z �U + |∇z �U|2 − D∆z �U − D∇z · �F = 0 , +(37) +with asymptotic BC �U → ∞ as |z| → ∞, or the reflecting +BC ( �F + ∇z �U) · �n = 0 on ∂�Ω, where �n denotes the unit +outer normal on ∂�Ω. We will only consider the rectangu- +lar domain case here. The argument for the unbounded +case is similar. +Recall that pss(x) satisfies the FPE +∇ · (pssF ) − D∆pss = 0. +(38) +Integrating both sides of (38) on Σ with respect to y +and utilizing the boundary condition Jss · n = 0, where +Jss = pssF − D∇pss, we get +∇z · +� � +Σ +Fzpss dy +� +− D∆z�pss = 0. +(39) +Taking (33) and (34) into account, we obtain +∇z · +� +�pss �F +� +− D∆z�pss = ∇z · � +J = 0, +(40) +i.e., a FPE for �pss(z) with the reduced force field �F , +where � +J := �pss �F −D∇z�pss. The corresponding boundary +condition can be also derived by integrating the original +BC Jss · n = 0 on Σ with respect to y for z ∈ ∂�Ω, which +gives +� +J · �n = +� +�pss �F − D∇z�pss +� +· �n = 0. +(41) +Substituting the relation �pss(z) = exp +� +−�U(z)/D +� +into +(40) and (41), we get (37) and the corresponding reflect- +ing BC after some algebraic manipulations. +E. +STATE-DEPENDENT DIFFUSION +COEFFICIENTS +In this section, we study the EPR loss for NESS sys- +tems with a state-dependent diffusion coefficient. +Consider the Ito SDEs +dx(t) +dt += F (x(t)) + +√ +2Dσ(x(t)) ˙w +(42) +with the state-dependent diffusion matrix σ(x). Under +the same assumptions as in the MT, we have the FPE +∇ · (pssF ) − D∇2 : (pssa) = 0. +(43) +We show that the high dimensional landscape function +U of (42) minimizes the EPR loss +LV-EPR(V ) = +� +Ω +|F v(x) + a(x)∇V (x)|2 +a−1(x) dπ(x), +(44) + +10 +where F v(x) := F (x) − D∇ · a(x) and |u|2 +a−1(x) := +u⊤a−1(x)u for u ∈ Rd. +To justify (44), we first note that (43) can be rewritten +as +∇ · (pssF v − Da∇pss) = 0 , +(45) +which, together with the BC, implies the orthogonality +relation +� +Ω +� +F v + a∇U +� +· ∇W dπ = 0 +(46) +for a suitable test function W(x). Following the same +reasoning used in establishing (18) and utilizing (46), we +have +� +Ω +|F v + a∇V |2 +a−1 dπ += +� +Ω +��F v + a∇U + a∇(V − U) +��2 +a−1 dπ += +� +Ω +|F v + a∇U +��2 +a−1 dπ + +� +Ω +��a∇(V − U) +��2 +a−1 dπ. +The last expression implies that U(x) is the unique min- +imizer of LV-EPR(V ) up to a constant. +The above derivation for the state-dependent diffusion +case will permit us to construct the landscape for the +chemical Langevin dynamics, which will be studied in +future work. +PART 2: COMPUTATION +Now we present the computational details and results +omitted in the MT in the computation part. +F. +2D MODELS AND COMPARISONS +In this section, we will describe the computational +setup and results for some 2D models which we utilize +for the test of different formulations, including the toy +model with known potential in the MT, a 2D biologi- +cal system with a limit cycle [3] and a 2D multi-stable +system [5]. We will also demonstrate the motivation for +enhanced EPR and its advantage over other methods. +F.1. +Toy model and enhanced EPR +In the toy model, we set the force field as +F (x) = −(I + A) · ∇U0(x), +(47) +and choose the potential +U0 = ((x − 1.5)2 − 1.0))2 + 0.5(y − 1.5)2, +(48) +where x = (x, y)⊤. We take the anti-symmetric matrix +A = +� +0 +0.5 +−0.5 +0 +� +, +(49) +which introduces a counter-clockwise rotation for a fo- +cusing central force field. +This sets up a simple non- +equilibrium system. In this model, we have +F (x) = −∇U0(x) + l(x), l(x) = −A · ∇U0(x) +and +l(x) · ∇U0(x) = 0 +holds in the pointwise sense. So, we have constructed +a double-well non-reversible system with analytically +known potential which can be used to verify the accu- +racy of the learned potential. We focus on the domain +Ω = [0, 3] × [0, 3]. +Primarily, the single EPR loss works well for the toy +model with a relatively large diffusion coefficient D = 0.1, +as shown in Fig. 1(A) in the MT. A slice plot of the poten- +tial at y = 1.5 (Fig. 3(A)) shows the EPR solution coin- +cides well with the analytical solution. The relative root +mean square error (rRMSE) and the relative mean abso- +lute error (rMAE), which will be defined in Section F F.4, +have mean and standard deviation of 0.099 ± 0.010 and +0.081 ± 0.013 over 3 runs, respectively. +However, when decreasing D to 0.05, the samples from +simulated invariant distribution mainly stay in the dou- +ble wells and away from the transition region (orange + +11 +FIG. 3. An illustration for the motivation of enhanced EPR. (A) and (B) show the comparisons of the learned potentials and +true solution on the line y = 1.5 in the toy model with D = 0.1 and D = 0.05, respectively. (C) shows the filled contour plot +of the potential learned by only the EPR loss. The orange points are samples from the simulated invariant distribution with +D = 0.05, While green points are enhanced samples simulated from a more diffusive distribution with D′ = 0.1, which are used +in the enhanced EPR. +points in Fig. 3(C)). In this case, the double well do- +main can still be learned well, yet the transition region, +without enough samples, has not been effectively trained. +Thus, as shown in Fig. 3(B), the single EPR result cap- +tures the double wells, but cannot accurately connect +them in the transition domain, which makes the left well +a bit higher than the right one. The pointwise HJB loss +with enhanced samples that better cover the transition +domain thus helps the EPR loss with samples for small +D, which mainly focuses on the local well domain. Us- +ing these enhanced samples for D′ = 0.1 (green points +in Fig. 3(A)), the enhanced EPR method performs much +better in the transition domain between the two wells +and thus agrees well with the true solution. +The above strategy is general, and we apply it to com- +pute the landscape for all of the continued 2D problems +and compare it with other methods in Section F F.4. +F.2. +2D limit cycle model +We apply our approach to the limit cycle dynamics +with a Mexican-hat shape landscape [3]. +Before proceeding to the concrete dynamical model, +we have the following observation. For any SDEs like +dx +dt = F (x) + +√ +2D ˙w, +(50) +the corresponding steady FPE is +∇ · (F pss) − D∆pss = 0. +If we make the transformation +F → κF , D → κD +in (50), then the steady state PDF +pss(x) ∝ exp +� +−U(x) +D +� += exp +� +−κU(x) +κD +� +is not changed. +The transformation only changes the +timescale of the dynamics (50) from t0 to κt0. However, +this transformation changes the learned potential from U +to κU if we utilize the drift κF (x) and noise strength κD +in the system (50), which is helpful to set the scale of U +to be O(1) by adjusting κ suitably for a specific problem. +An alternative approach to accomplish this task is by +choosing F to be κF in the EPR loss. +We take D = 0.1 and consider the limit cycle dynamics +dx +dt = κ +�α2 + x2 +1 + x2 +1 +1 + y − ax +� +, +(51) +dy +dt = κ +τ0 +� +b − +y +1 + cx2 +� +, +(52) +where the parameters are κ = 100, α = a = b = 0.1, c = +100, and τ0 = 5. Here the choice of κ = 100 is to make +U ∼ O(1) following [17]. We focus on the domain Ω = +[0, 8] × [0, 8] and compute the potential landscape and +force decomposition which is presented in the MT. As +explained in the above paragraph, this corresponds to +the case D = 0.1/κ = 0.001 for the force field considered +in [3]. + +B +A +C +1.4 +1.4 +3.0 +EPR +EPR +True +1.2 +1.2 - +True +一 +2.5 +Enhanced EPR +1.0 +1.0 +2.0 +0.8 - +0.8 +> +y1.5 +> +0.6 +0.6 +1.0 +0.4 +0.4 +0.5 +0.2 +0.2 +0.0 +0.0 +0.0 - +2.5 +0.5 +1.5 +0.5 +1.0 +1.5 +2.0 +1.0 +1.5 +2.0 +2.5 +2.5 +0.5 +1.0 +2.0 +3.0 +0.0 +3.0 +0.0 +3.0 +0.0 +X ++ ++12 +FIG. 4. Filled contour plots of the potential V (x; θ) for the toy model with D = 0.05 learned by (A) Enhanced EPR, (B) Naive +HJB, and (C) Normalizing Flow. The force field F (x) is decomposed into the gradient part −∇V (x; θ) (white arrows) and the +non-gradient part (gray arrows). The length of an arrow denotes the scale of the vector. The solid dots are samples from the +simulated invariant distribution. +F.3. +2D multi-stable model +We also apply the enhanced approach to study the +dynamics of a multi-stable system [5] +dx +dt = +axn +Sn + xn + +bSn +Sn + yn − k1x, +(53) +dy +dt = +ayn +Sn + yn + +bSn +Sn + xn − k2y, +(54) +where the parameters are a = b = k1 = k2 = 1, S = 0.5, +and n = 4. We focus on the domain Ω = [0, 3] × [0, 3] +and present the results for D = 0.01 in the MT. +F.4. +Numerical comparisons +In this subsection, we conduct a comparison study on +the previous 2D problems to show the priority of our +enhanced EPR approach over other methods. +For the +toy model, we have the analytical solution; while for the +other two 2D examples, we take the reference solution as +the numerical solution of the steady FPE by a piecewise +bilinear finite element method with fine rectangular grids +and the least squares solver for the obtained sparse lin- +ear system (a normalization condition +� +Ω pss(x)dx = 1 is +added to fix the extra shifting degree of freedom). +We use a fully connected neural network with 3 lay- +ers and 20 hidden states as the potential V (x; θ). We +train the network with a batch size of 2048 and a learn- +ing rate of 0.001 by the Adam [35] optimizer for 3000 +epochs. We simulate the SDEs by the Euler-Maruyama +scheme with reflecting boundaries on the boundary of +the domain and obtain a dataset of size 10000 to approx- +imate the invariant distribution. We update the dataset +by one time step at each training iteration to make it +closer to the invariant distribution. +In the toy model, +we try different scales to enhance samples and report the +best performance (when D′ = 2D) for naive HJB. For +fairness, we use the same enhanced samples in enhanced +EPR as naive HJB does. In SM, we denote the enhanced +loss as λ1 LEPR +λ2 LHJB and use λ1 = 0.1, λ2 = 1.0 in +the three models. We can also use Gaussian disturbances +of the SDE data to obtain enhanced data, as we do in +the limit cycle problem. We use D′ = 5D in the multi- +stable problem for a better covering of the transition do- +main. +For the comparison with normalizing flows, we +train a neural spline flow [36] using the implementation +from [37]. We repeat 4 blocks of the rational quadratic +spline with 3 layers of 64 hidden units and a followed LU +linear permutation. We train the flow model by Adam of +the learning rate 0.0001 for 20000 epochs, based on the +same sample dataset as enhanced EPR. +We shift the potential to the origin by its minimum +and focus on the domain +D = {x ∈ Ω|V (x; θ) ≤ 20D}. +We then define the modified potential +U m +0 (x) := min(U0(x), 20D), +V m(x; θ) := min(V (x; θ), 20D) +for the shifted potential U0(x) and V (x; θ) since only the +potential values in the domain D is of practical interest. +We use the relative root mean square error (rRMSE) and +the relative mean absolute error (rMAE) to describe the +accuracy. +rRMSE = +�� +Ω |V m(x; θ) − U m +0 (x)|2 dx +� +Ω |U m +0 (x)|2dx +, +(55) +rMAE = +� +Ω |V m(x; θ) − U m +0 (x)| dx +� +Ω |U m +0 (x)|dx +. +(56) +We summarize the comparison of numerical errors for +the 2D problems in Table I. The advantages of enhanced +EPR over both naive HJB and normalizing flow can be +identified from the following points. + +A +B +C +3.0 +3.0 +3.0 +1.4 +1.4 +1.4 +2.5 +2.5 +2.5 +1.2 +1.2 +1.2 +2.0 +2.0 +2.0 +1.0 +1.0 +1.0 +0.8 +0.8 +0.8 +1.5 +1.5 +>1.5 +0.6 +0.6 +0.6 +1.0 +1.0 +1.0 +0.4 +0.4 +0.4 +0.5 +0.5 +0.5 +0.2 +0.2 +0.2 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +1.5 +0.5 +1.0 +2.5 +0.5 +1.0 +1.5 +2.0 +2.5 +0.5 +1.5 +2.0 +3.0 +0.0 +0.0 +2.0 +2.5 +0.0 +3.0 +1.0 +3.0 +X +X +X13 +FIG. 5. Slices of the learned 3D potential V (x; θ) in the Lorenz system. The solid dots are samples from the simulated invariant +distribution. +TABLE I. Comparisons on Numerical Methods. We report +the mean and the standard deviation over 3 random seeds. +Problem +Method +rRMSE +rMAE +Toy, D=0.1 +Enhanced EPR +0.027±0.012 0.023±0.011 +Naive HJB +0.195±0.007 0.094±0.020 +Normalizing Flow 0.260±0.007 0.222±0.010 +Toy, D=0.05 +Enhanced EPR +0.048±0.021 0.030±0.012 +Naive HJB +0.237±0.020 0.142±0.042 +Normalizing Flow 0.284±0.028 0.231±0.030 +Limit Cycle +Enhanced EPR +0.052±0.039 0.029±0.016 +Naive HJB +0.107±0.043 0.048±0.019 +Normalizing Flow 0.255±0.007 0.210±0.015 +Multi-stable +Enhanced EPR +0.040±0.008 0.022±0.005 +Naive HJB +0.103±0.014 0.053±0.006 +Normalizing Flow 0.199±0.059 0.123±0.055 +• Without the guidance of EPR loss, naive HJB can +not effectively optimize to the true solution with +the heuristically chosen distribution. As shown in +Table I, the enhanced EPR significantly achieves +much better performances than naive HJB. Also, +in the toy model with D = 0.05, naively training +by HJB leads to an unreliable solution in Fig. 4(B) +with relative errors larger than 0.1. Our computa- +tional experiences show that the enhanced EPR is +more robust than naive HJB and less sensitive to +the enhanced data distribution and parameters. +• The enhanced EPR converges faster than the naive +HJB. For instance, in the toy model with D = 0.1, +the enhanced EPR has achieved rRMSE of 0.087 ± +0.069 and rMAE of 0.066 ± 0.013 in 2000 epochs, +while the naive HJB can not attain the same level +even after 3000 epochs. +• Without information from the dynamics, the nor- +malizing flow performs the worst only based on +the simulated invariant distribution dataset. The +learned potential tends to be rough and non- +smooth at the edge of samples as shown in Fig. 4. +Thus the enhanced EPR explicitly utilizing the in- +formation of the force field does help in more accu- +rate training of the potential. +We further compare the potential landscape computed +by different methods in Fig. 4. We remark that we omit +the space {x|V (x) ≥ 30D} in both Fig. 3 and Fig. 4 since +these domains are not of practical interest (their proba- +bility is less than 10−9 according to the Gibbs form of +the invariant distribution). The enhanced EPR presents +the landscape more consistent with the simulated sam- +ples and the true/reference solution than other methods. +The decomposition of the force also shows better match- +ing for the toy model. The normalizing flow captures the +high probability domain but lacks information on the dy- +namics, thus making its error larger than enhanced EPR +and naive HJB. +G. +3D MODELS +In this section, we describe the computational setup +for the Lorenz system in three dimensions and Ferrell’s +three-ODE model. We demonstrate the slices of the 3D + +20.0 +40 +17.5 +35 +15.0 +30 +12.5 +25 +Z +10.0 +20 +7.5 +15 +5.0 +2.5 +10 +0.0 +-15 -10 -5 +12 1416 +0 +0 +2 +5 +10 +4 +6 +X +1514 +FIG. 6. Streamlines of the projected force ˜ +G(z) and filled contour plot of the reduced potential ˜V (z; θ) for Ferrell’s three-ODE +model learned by enhanced EPR. +potential for the former and conduct the proposed di- +mensionality reduction on the latter. +G.1. +3D Lorenz system +In this subsection, we apply our landscape construction +approach to the 3D Lorenz system [38] with isotropic +temporal Gaussian white noise. +The Lorenz system has the form +dx +dt = β1(y − x), +(57) +dy +dt = x (β2 − z) − y, +(58) +dz +dt = xy − β3z, +(59) +where β1 = 10, β2 = 28 and β3 = 8 +3. We add the noise +with strength D = 1. This model was also considered +in [18] with D = 20. +We obtain the enhanced data by adding Gaussian +noises with standard deviation σ = 5 to the SDEs- +simulation data. +We directly train the 3D potential +V (x; θ) by enhanced EPR with λ1 = 10.0, λ2 = 1.0 +and present a slice view of the landscape in Fig. 5. The +learned 3D potential agrees well with the simulated sam- +ples and shows a butterfly-like shape as the original sys- +tem does. +G.2. +Ferrell’s three-ODE model +In this subsection, we consider Ferrell’s three-ODE +model for a simplified cell cycle dynamics [29] denoted +by +x = [CDK1], y = [Plk1], z = [APC] + +0.7 +0.200 +0.175 +0.6 +0.150 +0.5 +0.125 +0.4 +Plk1 +0.100 +0.3 +0.075 +0.2 +0.050 +0.1 +0.025 +0.0 +0.000 +0.0 +0.3 +0.5 +0.1 +0.2 +0.6 +0.4 +0.7 +CDK115 +for the concentration of CDK1, Plk1, and APC. We have +the ODEs +dx +dt = α1 − β1x +zn1 +Kn1 +1 ++ zn1 , +(60) +dy +dt = α2 (1 − y) +xn2 +Kn2 +2 ++ xn2 − β2y, +(61) +dz +dt = α3 (1 − z) +yn3 +Kn3 +3 ++ yn3 − β3z, +(62) +where α1 = 0.1, α2 = α3 = β1 = 3, β2 = β3 = 1, K1 = +K2 = K3 = 0.5, n1 = n2 = 8, and n3 = 8. We add the +noise scale D = 0.01 with isotropic temporal Gaussian +white noise. +By taking the reduced variables z = (x, y)⊤, we can +apply our force projection loss and enhanced loss to learn +the projected force ˜G(x) and potential ˜V (x; θ), and the +results are shown in Fig. 6. +We use three-layer net- +works with 80 hidden states in this problem and en- +hanced samples simulated from a more diffusive distribu- +tion with D′ = 5D. We train the projected force ˜G(x) +for 1000 epochs and then conduct enhanced EPR with +λ1 = 0.1, λ2 = 1.0 for 4000 epochs to compute the pro- +jected potential. The obtained reduced potential shows +a plateau in the centering region and a local-well tube +domain along the reduced limit cycle. +H. +HIGH DIMENSIONAL MODELS +In this section, we apply our approach to 8D limit cy- +cle dynamics [4] and 52D multistable dynamics [39]. We +directly train the reduced force field ˜G(z) and poten- +tial ˜V (z; θ) according to the selected reduction variables +suggested in the corresponding literature. We use three- +layer networks with 80 hidden states for both force and +potential. The training strategies are similar to previous +examples. +H.1. +8D complex system +We consider an 8D system in which the dynamics and +parameters are the same as the supporting information +of [4], and take CycB and Cyc20 as the reduction variable +z. We set the mass in this problem as m = 0.8. +In [4], the noise strength D = 0.0005 is not suitable for +direct neural network training since the scale of the po- +tential is O(10−5). Borrowing the idea in Section F F.2, +we amplify the original force field F considered in [4] +by κ = 1000 times, and take D = 0.01 for the trans- +formed force field. This amounts to set D = 10−5 for the +original force field, which is even smaller than the case +considered in [4]. We simulate the SDEs without bound- +aries first and then fix the dataset without updating. We +obtain the enhanced samples by adding Gaussian pertur- +bations to the obtained dataset. Only the data within the +FIG. 7. Streamlines and limit sets of the projected force field +of the 8D cell cycle model by two reduced variables CycB and +Cdc20. The outer red circle is the stable limit cycle of the +reduced force field corresponding to the yellow circle as the +projection of the original high dimensional limit cycle. The +inner red circle, red dot and two green circles are stable and +unstable limit sets of the reduced dynamics, which are virtual +in high dimensions. +biologically meaningful domain of [0, 1.5]8 is utilized for +computation. +We train the projected force ˜G(z; θ) for 5000 epochs +and conduct the enhanced EPR with λ1 = 0.1, λ2 = 1.0 +for 10000 epochs. Some essential features of the reduced +potential and dynamics on the plane have been presented +in MT. +In the SM Fig. 7, we present a more thorough picture +of the reduced dynamics for the 8D model than the MT +Fig. 2. To be more specific, we further show two unstable + +0.8 +0.6 +Cdc20 +0.4 +0.2 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +CycB16 +FIG. 8. Projected force ˜ +G(x) and potential ˜V (x; θ) of the 52D double-well model learned by enhanced EPR. +limit cycles of the projected force field, two green circles +obtained by reverse time integration, in SM Fig. 7. They +fall between the outer and inner stable limit cycles (inner +and outer red circles), and the inner stable limit cycle and +inner stable node (red dot in the center), which play the +role of separatrices between the neighboring stable limit +sets. This picture occurs as the result that the landscape +of the considered system in the centering region is very +flat. These inner limit sets are virtual in high dimensions, +but they naturally appear in the reduced dynamics on the +plane. Similar features might also occur in other reduced +dynamics in two dimensions. +H.2. +52D multi-stable system +We also apply our approach to a biological system +with 52 ODEs constructed by [39] and take GATA6 and +NANOG as the reduction variable z. We define Ai as +the set of indices for activating xi and Ri as the set of +indices for repressing xi, the corresponding relationships +are defined as the 52D node network shown in [39]. For +i = 1, ..., 52, +dxi +dt = −kxi + +� +j∈Aj +axn +j +Sn + xn +j ++ +� +j∈Rj +bSn +Sn + xn +j +, +(63) +where a = 0.37, b = 0.5, k = 1, S = 0.5, and n = 3. We +choose the noise strength D = 0.01. +We train the force ˜G(z; θ) for 500 epochs and conduct +enhanced EPR with λ1 = 100.0, λ2 = 1.0 for 500 epochs. +We use enhanced samples simulated from a more diffusive +distribution with D′ = 5D. +As shown in Fig. 8, the +projected force demonstrate the reduced dynamics and +the depth of the constructed potential agrees well with +the density of the sample points. + +2.00 +0.200 +1.75 +0.175 +1.50 +0.150 +1.25 +0.125 +IANOG +1.00 +0.100 +0.75 +0.075 +0.050 +0.50 +0.025 +0.25 +0.000 +0.00 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +GATA617 +[1] C. Waddington, The Strategy of the Genes (George Allen +& Unwin, Ltd., London, 1957). +[2] P. Ao, J. Phys. A-Math. Gen. 37, L25 (2004). +[3] J. Wang, L. Xu, and E. Wang, Proc. Nat. Acad. Sci. USA +105, 12271 (2008). +[4] J. Wang, C. Li, and E. Wang, Proc. Nat. Acad. Sci. USA +107, 8195 (2010). +[5] J. Wang, K. Zhang, L. Xu, and E. Wang, Proc. Nat. +Acad. Sci. USA 108, 8257 (2011). +[6] J. Zhou, M. Aliyu, E. 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Biol. 9, e1003165 +(2013). + diff --git a/-dAzT4oBgHgl3EQf_P7B/content/tmp_files/load_file.txt b/-dAzT4oBgHgl3EQf_P7B/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ede206d4704fbafea1c835375037e21b5fc44530 --- /dev/null +++ b/-dAzT4oBgHgl3EQf_P7B/content/tmp_files/load_file.txt @@ -0,0 +1,1166 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf,len=1165 +page_content='EPR-Net: Constructing non-equilibrium potential landscape via a variational force projection formulation Yue Zhao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1 Wei Zhang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' ∗ and Tiejun Li1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' † 1Center for Data Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Beijing 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' China 2Zuse Institute Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' D-14195 Berlin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Germany 3LMAM and School of Mathematical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Beijing 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' China 4Center for Machine Learning Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Peking University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Beijing 100871,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' China (Dated: January 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 2023) We present a novel yet simple deep learning approach,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' dubbed EPR-Net,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' for constructing the potential landscape of high-dimensional non-equilibrium steady state (NESS) systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The key idea of our approach is to utilize the fact that the negative potential gradient is the orthogonal projection of the driving force in a weighted Hilbert space with respect to the steady-state distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The constructed loss function also coincides with the entropy production rate (EPR) formula in NESS theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' This approach can be extended to dealing with dimensionality reduction and state-dependent diffusion coefficients in a unified fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The robustness and effectiveness of the proposed approach are demonstrated by numerical studies of several high-dimensional biophysical models with multi- stability, limit cycle, or strange attractor with non-vanishing noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Since Waddington’s famous landscape metaphor on the development of cells in the 1950s [1], the construction of potential landscape for non-equilibrium biochemical reac- tion systems has been recognized as an important prob- lem in theoretical biology, as it provides insightful pic- tures for understanding complex dynamical mechanisms of biological processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' This problem has attracted con- siderable attention in recent decades in both biophysics and applied mathematics community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Until now, several approaches have been proposed to realize Waddington’s landscape metaphor in a rational way, see [2–10] and the references therein for details and [11–14] for reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Broadly speaking, these proposals can be classified into two types: (T1) the construction of potential landscape in the finite noise regime [3–5] and (T2) the construction of the quasi-potential in the zero noise limit [2, 6–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' For low-dimensional systems (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=', dimension less than 4), the potential landscape can be numerically computed either by solving a Fokker-Planck equation (FPE) using grid-based methods until the steady solution is reached approximately as in (T1) type proposals [3, 5], or by solv- ing a Hamilton-Jacobi-Bellman (HJB) equation using, for instance, the ordered upwind method [15] or minimum action type method [8] as in (T2) type proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' How- ever, these approaches suffer from the curse of dimen- sionality when applied to high-dimensional systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Al- though methods based on mean field approximations are able to provide a semi-quantitative description of the en- ergy landscape for typical systems [4, 16], direct and gen- eral approaches are still favored in applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In this aspect, pioneering work has been done recently, which allows direct construction of high-dimensional potential landscape using deep neural networks (DNN), based on either the steady viscous HJB equation satisfied by the ∗ wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='zhang@fu-berlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='de † tieli@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='cn landscape function in (T1) case [17, 18], or the point- wise orthogonal decomposition of the force field in (T2) case [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' These works have brought significant advances in the methodological developments in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' How- ever, these approaches, which are based on solving HJB equations alone, may encounter numerical difficulties due to the non-uniqueness of the weak solution to the non- viscous HJB equation in (T2) case [20], and challenges in solving the steady HJB equation with a small noise in (T1) case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In this letter, we present a simple yet ef- fective DNN approach, EPR-Net, for constructing the potential landscape of high-dimensional non-equilibrium steady state (NESS) systems in (T1) type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Our key ob- servation is that the negative potential gradient is the or- thogonal projection of the driving force under a weighted inner product with respect to the steady-state distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' To be specific, let us consider the stochastic differ- ential equations (SDEs) dx(t) dt = F (x(t)) + √ 2D ˙w, x(0) = x0, (1) where x0 ∈ Rd, F : Rd → Rd is a smooth function, ˙w = ( ˙w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' , ˙wd)⊤ is the d-dimensional temporal Gaus- sian white noise with E ˙wi(t) = 0 and E[ ˙wi(t) ˙wj(s)] = δijδ(t − s) for i, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' , d, s, t > 0 and D > 0 is the noise strength, which is often related to the system’s tem- perature T by D = kBT, where kB is the Boltzmann con- stant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We assume that (1) is ergodic and denote by pss(x) its steady-state probability density function (PDF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We follow the (T1) type proposal in [3] to derive the po- tential landscape of (1) in the case of D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' That is, we define the potential U = −D ln pss and the steady proba- bility flux Jss = pssF −D∇pss in the domain Ω, which we assume for simplicity is either Rd or a d-dimensional hy- perrectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The steady-state PDF pss(x) satisfies the Fokker-Planck equation (FPE) ∇ · (pssF ) − D∆pss = 0, for x ∈ Ω, (2) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='01946v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='bio-ph] 5 Jan 2023 2 and we assume the asymptotic boundary condition (BC) pss(x) → 0 as |x| → ∞ when Ω = Rd, or the re- flecting boundary condition Jss · n = 0 when Ω ⊂ Rd is a d-dimensional hyperrectangle, where n is the unit outer normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In both cases, we have pss(x) ≥ 0 and � Ω pss(x) dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Aiming at an effective approach for high-dimensional applications, we employ DNNs to approximate U(x), and the key idea in this letter is to learn U by training DNN with the following loss function LEPR(V ) = � Ω |F (x) + ∇V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ)|2 dπ(x), (3) where V := V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) is a neural network function with parameters θ [21], and dπ(x) = pss(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' To justify (3), we note that U satisfies the important orthogonality relation: for any suitable function W : Rd → R, � Ω � F (x) + ∇U(x) � ∇W(x) dπ(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (4) Therefore, U(x) is the unique minimizer (up to a con- stant) of the loss LEPR and, moreover, the negative po- tential gradient −∇U is in fact the projection of the force field F in the π-weighted Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' See Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' A and B in the Supplemental Material (SM) for derivations in de- tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The minimum loss LEPR(U) has a clear physical inter- pretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Indeed, we have (see SM Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' B) LEPR(U) = � Ω |Jss|2 1 pss dx = ess p , (5) where ess p denotes the steady entropy production rate (EPR) of the NESS system (1) [3, 22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Therefore, minimizing (3) is equivalent to approximating the steady EPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' This explains the name EPR-Net of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' To utilize (3) in numerical computations, we replace the spatial integral in (3) with respect to the unknown π by its empirical average using data sampled from (1): �LEPR(θ) = 1 N N � i=1 ��F (xi) + ∇V (xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) ��2, (6) where (xi)1≤i≤N could be either the final states (at time T) of N trajectories starting from different initializations or equally spaced time series along a single long trajec- tory up to time T, where T ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In both cases, the ergodicity of SDE (1) guarantees that (6) is a good ap- proximation of (3) as long as T is large [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We adopt the former approach in the numerical experiments in this work, where the gradients of both V (with respect to x) and the loss itself (with respect to θ) in (6) are calculated by auto-differentiation through PyTorch [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The stabil- ity analysis of this approximation is presented in detail in SM Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We apply our method to a toy model first in order to check its applicability and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We take F (x) = −(I + A) · ∇U0(x), (7) where A ∈ Rd×d is a constant skew-symmetric matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=', A⊤ = −A, and U0 is some known function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' With this choice of F , one can check that the true potential land- scape is U(x) = U0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In particular, the system is re- versible when A = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Based on the proposed method, we construct a double-well model with known potential U0 for verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We take D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 1(A), the learned potential agrees well with the simulated sam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Also, the decomposition of the force field shows that the negative gradient part −∇V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) around the wells points towards the attractor and is nearly orthog- onal to the non-gradient part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The overall non-gradient field shows a counter-clockwise rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The relative root mean square error (rRMSE) of the potential V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) learned by EPR loss is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0987 (averaged over 3 runs), which supports the effectiveness of our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' See SM Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' F F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1 for details of the problem setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The correct interpretation of the computational results based on the EPR loss (3) is that the accuracy of V (x) is guaranteed only when π(x) is evidently above zero for any specific x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In the “visible” domain of π (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=', the places where there are sample points of {xi}), the trained potential V gives reliable approximation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' while in the weakly visible or invisible domain, especially in local transition regions between meta-stable states and boundaries of the visible domain, we must resort to the original FPE (2) which holds pointwise in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Learning strategy for small D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Substituting the rela- tion pss(x) = exp(−U(x)/D) into (2), we get the viscous HJB equation NHJB(U) := F · ∇U + |∇U|2 − D∆U − D∇ · F = 0 (8) with the asymptotic BC U → ∞ as |x| → ∞ in the case of Ω = Rd, or the reflecting BC (F + ∇U) · n = 0 on ∂Ω when Ω is a d-dimensional hyperrectangle, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' As in the framework of physics-informed neural networks (PINNs) [26], (8) motivates the HJB loss LHJB(V ) = � Ω ��NHJB(V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ)) ��2 dµ(x), (9) where µ is any desirable distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' By choosing µ properly, this loss allows the use of sample data that better cover the domain Ω and, when combined with the loss in (3), leads to significant improvement of the train- ing results in our numerical experiments when D is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Specifically, for small D, we propose the enhanced loss in training which has the form �Lenh(θ) = �LEPR(θ) + λ �LHJB(θ), (10) where �LEPR(θ) is defined in (6), �LHJB(θ) = 1 N ′ �N ′ i=1 |NHJB(V (x′ i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ))|2 is an approximation of (9) us- ing an independent data set (x′ i)1≤i≤N ′ obtained by sam- pling the trajectories of (1) with a larger D′ > D, and λ > 0 is a weight parameter balancing the contribution of the two terms in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Note that the proposed strategy is both general and easily adaptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' For instance, one 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Filled contour plots of the learned potential V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) for (A) toy model learned by EPR loss (3) with D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1, and (B)-(C) a biochemical oscillation network model [3] and a tri-stable cell development model [5] learned by enhanced loss (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The force field F (x) is decomposed into the gradient part −∇V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) (white arrows) and the non-gradient part (gray arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The length of an arrow denotes the scale of the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The solid dots are samples from the simulated invariant distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' can alternatively use data (x′ i)1≤i≤N ′ that contains more samples in the transition region, or employ a modification of the loss (9) in (10) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We apply our enhanced loss (10) to construct the land- scape for a 2D biological system with a limit cycle [3] and a 2D multistable system [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The potential V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) learned by the enhanced loss (10), the force decomposi- tion, and sample points from the simulated invariant dis- tribution are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 1(B) and (C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' As in the toy model case, the gradient part (white arrows) points di- rectly towards the attractors, while the non-gradient part (gray arrows) shows a counter-clockwise rotation for the limit cycle, and a splitting-and-back flow from the mid- dle attractor to the other two attractors for the tri-stable dynamical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' To further verify the accuracy of the method, we numerically solve the FPE (2) as reference solutions by a fine grid discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Comparisons be- tween the proposed method and the method based on the naive HJB loss on these two problems are demon- strated in SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Averaged over 3 runs, the rRMSE of the potential V learned by our enhanced loss is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0524 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0402, respectively, which shows an evident advantage over the naive HJB loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' See SM Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' F for details of the comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Dimensionality reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' When applying the ap- proach above to high-dimensional problems, dimensional- ity reduction is necessary in order to visualize the results and gain physical insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' A straightforward approach is to first learn the high-dimensional potential U and then find its low-dimensional representation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=', the reduced potential or the free energy function, using dimension- ality reduction techniques (see SM Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' D D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In the following, we present an alternative approach that allows to directly learn the low-dimensional reduced potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' For simplicity, we consider the linear case and, with a slight abuse of notation, denote by x = (y, z)⊤, where z = (xi, xj) ∈ R2 contains the coordinates of two vari- ables of interest, and y ∈ Rd−2 corresponds to the other d − 2 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The domain Ω (either Rd or a d-dimensional hyperrectangle) has the decomposition Ω = Σ × �Ω, where Σ ⊆ Rd−2 and �Ω ⊆ R2 are the do- mains of y and z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' As can be seen in the numerical examples, this setting is applicable to many interesting biochemical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Extensions to nonlinear low-dimensional reduced variables with general domains are possible, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=', by applying the approach developed in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In the current setting, the reduced potential is �U(z) = −D ln �pss(z) = −D ln � Σ pss(y, z) dy, (11) and one can show that �U minimizes the following loss function: LP-EPR(�V ) = � Ω ��Fz(y, z)+∇z �V (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) ��2 dπ(y, z), (12) where Fz(y, z) ∈ R2 is the z-component of the force field F = (Fy, Fz)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Similar to (6), the empirical form of (12) can be used in learning the reduced potential �U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Moreover, one can derive an enhanced loss as in (10) that could be used for systems with small D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' To this end, we note that �U satisfies the projected HJB equation NP-HJB(�U) := �F · ∇z �U + |∇z �U|2 − D∆z �U − D∇z · �F = 0 , (13) with asymptotic BC �U → ∞ as |z| → ∞, or the reflecting BC ( �F + ∇z �U) · �n = 0 on ∂�Ω, where �F (z) := � Σ Fz(y, z)dπ(y|z) is the projected force defined using the conditional distribution dπ(y|z) = pss(y, z)/�pss(z) dy, and �n denotes the unit outer normal on ∂�Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Based on (13), we can formulate the projected HJB loss LP-HJB(�V ) = � �Ω ��NP-HJB(�V (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ)) ��2 dµ(z), (14) A B 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='20 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 2 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 0 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 + X X4 where µ is any suitable distribution over �Ω, and �F in (13) is learned beforehand by training a DNN with the loss LP-For( � G) = � Ω ��Fz(y, z) − � G(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) ��2 dπ(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (15) The overall enhanced loss used in numerical computa- tions comprises two terms, which are empirical estimates of (12) and (14) based on two different sets of sample data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' See SM Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' D for derivation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We then apply our dimensionality reduction approach to construct the landscape for an 8D cell cycle model con- taining both a limit cycle and a stable equilibrium point for the chosen parameters, and take CycB and Cdc20 as the reduced variables following [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 2, we can find that the depth of the reduced potential and force strength agree well with the density of projected samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Moreover, we can also get some important insights from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 2 on the projection of the high-dimensional dynam- ics with a limit cycle to two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' One particular feature is that the limit cycle induced by the projected force � G (outer red circle) has minor differences with the limit cycle directly projected from high dimensions (yel- low circle), and the difference is slight or moderate de- pending on whether the density of samples is high or low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' This is natural in the reduction since the distribu- tion π(y|z) in the projection is not of Dirac type when D > 0, and this difference will disappear as D → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Another feature is that we unexpectedly get an addi- tional stable limit cycle (inner red circle) and a stable point (red dot in the center) emerging inside the limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Though virtual in high dimensions and biologi- cally irrelevant, the existence of such two limit sets is reminiscent of the Poincar´e-Bendixson theorem in pla- nar dynamics theory [28, Chapter 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='6], which depicts a common phenomenon when performing dimensionality reduction with limit cycles to 2D plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The emergence of these two limit sets, though being not a general sit- uation, is specific in the considered model due to the relatively flat landscape of the potential in the centering region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In addition, close to the saddle point (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='13, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='55) of �V (green star), there is a barrier domain along the limit cycle direction, while a local well domain along the Cdc20 direction, which characterizes the region that bi- ological cycle paths mainly go through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Last but not the least, a zoom-in view of the local well domain out- side of the limit cycle shows its detailed spiral structure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 2C), which has not been revealed before by mak- ing a Gaussian approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Some other applications of our approach to Ferrell’s three-ODE model [29], 52D stem cell network model [16] and 3D Lorenz model are demonstrated in SM Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' G and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Extension to variable diffusion coefficient case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The EPR-Net formulation can be extended to the case of state-dependent diffusion coefficients without any diffi- culty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Consider the Ito SDEs dx(t) dt = F (x(t)) + √ 2Dσ(x(t)) ˙w, x(0) = x0, (16) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Dimensionality reduction of an 8D cell cycle model with two reduced variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (A) Reduced potential landscape �V with projected contour lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (B) Projected sample points, streamlines of the projected force field � G and the filled con- tour plot of �V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The red circles and dots are stable limit sets of the projected force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The yellow circle is the projection of the original high-dimensional limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (C) The detailed spiral structure of the streamlines of � G around the stable point by zooming in the square domain in (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' with diffusion matrix σ(x) ∈ Rd×m and ˙w is an m- dimensional temporal Gaussian white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We assume that m ≥ d and the matrix a(x) := (σσ⊤)(x) satisfies u⊤a(x)u ≥ c0|u|2 for all x, u ∈ Rd, where c0 > 0 is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Using a similar derivation as before, we can again show that the high-dimensional landscape function U of (16) minimizes the EPR loss LV-EPR(V ) = � Ω |F v(x) + a(x)∇V (x)|2 a−1(x) dπ(x), (17) where F v(x) = F (x) − D∇ · a(x) and |u|2 a−1(x) := u⊤a−1(x)u for u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We provide derivation details of (17) in SM Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' However, we will not pursue a nu- merical study of (16)–(17) in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Discussions and Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Below we make some fi- nal remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' First, concerning the use of the steady-state distribution π(x) in (3) and its approximation by a long time series of the SDE (1) in EPR-Net, we emphasize that it is the sampling approximation of π that naturally cap- tures the important parts of the potential function, and the landscape beyond the sampled regions is not that essential in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Second, as is exemplified in SM Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' F F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4, we found that a direct application of density estimation methods (DEM), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=', normalizing flows [30], to the sampled time series data does not give potential A B 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='08 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 CycB Cdc20 C 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='16 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 CycB5 landscape with satisfactory accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We speculate that such deficiency of DEM is due to its over-generality and the fact that it does not take advantage of the force field information explicitly compared to (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Overall, we have presented the EPR-Net, a simple yet effective DNN approach, for constructing the non- equilibrium potential landscape of NESS systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' This approach is both elegant and robust due to its variational structure and its flexibility to be combined with other types of loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Further extension of dimensional- ity reduction to nonlinear reduced variables and numeri- cal investigations in the case of state-dependents diffusion coefficients will be explored in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We thank Professors Chunhe Li, Xiaoliang Wan and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Yufei Ma for helpful discus- sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' TL and YZ acknowledge the support from NSFC and MSTC under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='s 11825102, 12288101 and 2021YFA1003300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' WZ is supported by the DFG un- der Germany’s Excellence Strategy-MATH+: The Berlin Mathematics Research Centre (EXC-2046/1)-project ID: 390685689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The numerical computations of this work were conducted on the High-performance Computing Platform of Peking University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 6 Supplemental Material for: EPR-Net: Constructing non-equilibrium potential landscape via a variational force projection formulation CONTENTS Part 1: Theory 6 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Validation of the EPR loss 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' EPR loss and entropy production rate 7 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Stability of the EPR minimizer 7 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Dimensionality reduction 8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Gradient projection loss 8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Projected EPR loss 8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Force projection loss 9 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' HJB equation for the reduced potential 9 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' State-dependent diffusion coefficients 9 Part 2: Computation 10 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 2D models and comparisons 10 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Toy model and enhanced EPR 10 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 2D limit cycle model 11 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 2D multi-stable model 12 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Numerical comparisons 12 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 3D models 13 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 3D Lorenz system 14 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Ferrell’s three-ODE model 14 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' High dimensional models 15 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 8D complex system 15 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 52D multi-stable system 16 References 17 In this supplemental material (SM), we will present further theoretical derivations and computational details of the contents in the main text (MT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' This SM consists of two parts: Theory and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' PART 1: THEORY We will first provide details of theoretical derivations omitted in the MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' VALIDATION OF THE EPR LOSS In this section, we show that, up to an additive con- stant, the potential function U(x) := −D ln pss(x) is the unique minimizer of the EPR loss (3) defined in the MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' First, we show that the orthogonality relation � Ω (F + ∇U) · ∇W dπ = 0 (18) holds for any suitable function W(x) : Rd → R under both choices of the boundary conditions (BC) considered in the MT, where dπ(x) := pss(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' To see this, we note that � Ω (F + ∇U) · ∇W dπ = � Ω (F pss − D∇pss) · ∇W dx = � ∂Ω W(F pss − D∇pss) · n dx − � Ω W∇ · (F pss − D∇pss) dx :=P1 − P2 where we have used integration by parts and the relation pss(x) = exp(−U(x)/D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The term P1 is zero due to the fact that pss(x) tends to 0 exponentially as |x| → ∞ when Ω = Rd, and the reflecting BC Jss · n = 0 which holds on ∂Ω when Ω is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The term P2 is zero due to the steady state Fokker-Planck equation (FPE) satisfied by pss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Now consider the EPR loss, we have LEPR(V ) = � Ω |F + ∇V |2 dπ = � Ω |F + ∇U + ∇V − ∇U|2 dπ = � Ω � |F + ∇U|2 + |∇V − ∇U|2� dπ + 2 � Ω (F + ∇U) · ∇(V − U) dπ = � Ω |F + ∇U|2 + |∇V − ∇U|2 dπ, where we have used the orthogonality relation (18) to arrive at the last equality, from which we conclude that 7 U(x) is the unique minimizer of the EPR loss up to an additive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In fact, define the π-weighted inner product for any square integrable functions f, g on Ω: (f, g)π := � Ω f(x)g(x) dπ(x) (19) and the corresponding L2 π-norm ∥·∥π by ∥f∥2 π := (f, f)π, we get a Hilbert space L2 π (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=', [31, Chapter II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Choosing W = U in (18), we observe that the minimiza- tion of EPR loss finds the orthogonal projection of F under the π-weighted inner product, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=', F (x) = −∇U(x) + l(x), such that (∇U, l)π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (20) However, we remark that this orthogonality holds only in the L2 π-inner product sense instead of the pointwise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Furthermore, the two orthogonality relations (18) and (20) can be understood as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Using (20), the relation (18) is equivalent to � Ω l · ∇Wdπ = 0 for any W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Integration by parts gives ∇ · (l e−U/D) = 0, which is equivalent to ∇U · l + D∇ · l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' When D → 0, we recover the pointwise orthogonality, which is adopted in computing quasi-potentials in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' EPR LOSS AND ENTROPY PRODUCTION RATE In this section, we show that the minimum EPR loss coincides with the steady entropy production rate in non- equilibrium steady state (NESS) theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Following [22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 23],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' we have the important identity con- cerning the entropy production for the SDE (1) defined in the MT: DdS(t) dt = ep(t) − hd(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (21) where S(t) := − � Ω p(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' t) ln p(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' t) dx is the entropy of the probability density function p(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' t) at time t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' ep is the entropy production rate (EPR) ep(t) = � Ω |F (x) − D∇ ln p(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' t)|2 p(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' t) dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (22) and hd is the heat dissipation rate hd(t) = � Ω F (x) · J(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' t) dx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (23) with the probability flux J(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' t) := p(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' t)(F (x) − D∇ ln p(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' t)) at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' When D = kBT, the above for- mulas have clear physical meaning in statistical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' At the steady state, we get the steady EPR ess p = � Ω |F − D∇ ln pss|2 pss dx = � Ω |F + ∇U|2 pss dx = � Ω |Jss|2 1 pss dx = LEPR(U), where Jss(x) = pss(x)(F (x)+∇U(x)) is the steady prob- ability flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' This shows the relation between the pro- posed EPR loss function and the entropy production rate in the NESS theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' STABILITY OF THE EPR MINIMIZER In this section, we formally show that small perturba- tions of the invariant distribution π will not introduce a disastrous change to the minimizer of the correspond- ing EPR loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We only consider the bounded domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=', Ω is a hyperrectangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The argument for unbounded domains is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Suppose dπ(x) = p(x)dx, dµ(x) = q(x)dx, and the functions U(x) and ¯U(x) are the unique minimizers (up to a constant) of the following two EPR losses U = arg min V � Ω |F + ∇V |2 dπ, ¯U = arg min V � Ω |F + ∇V |2 dµ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' It is not difficult to find that the Euler- Lagrange equations of U, ¯U are given by the following partial differential equation (PDE) with suitable BCs: ∇ · ((F + ∇U)p) = 0 in Ω, (F + ∇U) · n = 0 on ∂Ω, ∇ · ((F + ∇ ¯U)q) = 0 in Ω, (F + ∇ ¯U) · n = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The PDEs above defined inside the domain Ω can be converted to ∆Up + ∇U · ∇p = −∇ · (pF ), ∆ ¯Uq + ∇ ¯U · ∇q = −∇ · (qF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Define U0(x) = −D ln p(x) and ¯U0(x) = −D ln q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We then obtain −∇U · ∇U0 + D∆U = F · ∇U0 − D∇ · F , (24) −∇ ¯U · ∇ ¯U0 + D∆ ¯U = F · ∇ ¯U0 − D∇ · F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (25) Assuming that δU0 := U0 − ¯U0 = O(ε), where 0 < ϵ ≪ 1 denotes a small constant, we have the PDE for U − ¯U by subtracting (25) from (24): −∇(U− ¯U) · ∇U0 + D∆(U − ¯U) = F · ∇(δU0) + ∇ ¯U · ∇(δU0) with BC ∇(U − ¯U) · n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Since U0, ¯U, F ∼ O(1), we can obtain that U(x) − ¯U(x) = O(ε) by the regularity theory of elliptic PDE [32, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='3] when D ∼ O(1), or by the matched asymptotic expan- sion when D ≪ 1 [33, Chapter 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In fact, the closeness between U(x) and ¯U(x) can be ensured as long as U0 and ¯U0 are close enough in the region where p(x) and q(x) are bounded away from zero by the method of characteristics analysis [32, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1] and matched asymptotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 8 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' DIMENSIONALITY REDUCTION In this section, we study dimensionality reduction for high-dimensional problems in order to learn the projected potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Denote by x = (y, z)⊤ ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' As in the MT, we assume the domain Ω = �Ω × Σ, where �Ω ⊆ R2 and Σ ⊆ Rd−2 are the domain of y and z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The reduced potential �U(z) is defined as �U(z) = −D ln �pss(z) = −D ln � Σ pss(y, z) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (26) One natural approach for constructing �U(z) is directly integrating pss(y, z) based on the learned U(y, z) with the EPR loss, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=', �U(z) = −D ln � Σ exp(−U(y, z)/D) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (27) However, performing this integration is not a straightfor- ward numerical task (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=', [34, Chapter 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Gradient projection loss In this subsection, we study a simple approach to approximate �U(z) based on sample points, which ap- proximately obey the invariant distribution π(x), and the learned high dimensional potential function U(x) by EPR loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' This approach is not investigated numerically in this work, but it will be useful for the derivations in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The idea is to utilize the gradient projection (GP) loss on the z components of ∇U: LGP(�V ) = � Ω ��∇zU(y, z) − ∇z �V (z) ��2 dπ(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (28) To justify (28), we note that LGP(�V ) = � Ω ��∇zU − ∇z �V ��2 dπ(x) = � Ω ��∇zU − ∇z �U + ∇z �U − ∇z �V ��2 dπ(x) = � Ω ���∇zU − ∇z �U ��2 + ��∇z �U − ∇z �V ��2� dπ(x) + 2 � Ω (∇zU − ∇z �U) · ∇z(�U − �V ) dπ(x) =:P1 + P2, where P1 and P2 denote the terms in the third and the fourth line above, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The term P2 = 0 since � Ω ∇zU · ∇z(�U − �V ) dπ(x) = � �Ω �� Σ ∇zUe− U D dy � ∇z(�U − �V ) dz = − D � �Ω ∇z �� Σ e− U D dy � ∇z(�U − �V ) dz = − D � �Ω ∇z�pss · ∇z(�U − �V ) dz = � �Ω ∇z �U · ∇z(�U − �V ) �pss dz and � Ω ∇z �U · ∇z(�U − �V ) dπ(x) = � �Ω ∇z �U · ∇z(�U − �V ) �pss dz, which cancel with each other in P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Therefore, the minimization of GP loss is equivalent to minimizing � �Ω ��∇z �U − ∇z �V ��2 �pss dz, which clearly implies that �U(z) is the unique minimizer (up to a constant) of the proposed GP loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Projected EPR loss In this subsection, we study the projected EPR (P- EPR) loss, which has the form LP-EPR(�V ) = � Ω ��Fz(y, z) + ∇z �V (z) ��2 dπ(y, z), (29) where Fz(y, z) ∈ R2 is the z-component of the force field F = (Fy, Fz)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Define �LP-EPR(�V ) = � Ω ��F (y, z) + ∇�V (z) ��2 dπ(y, z), (30) where ∇ is the full gradient with respect to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' To justify (29), we first note the following equivalence min LP-EPR(�V ) ⇐⇒ min �LP-EPR(�V ), (31) since ∇y �V (z) = 0 and the y-components of F +∇�V only introduce an irrelevant constant in (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Furthermore, we have �LP-EPR(�V ) = � Ω ��F + ∇�V ��2 dπ(x) = � Ω ��F + ∇U + ∇�V − ∇U ��2 dπ(x) = � Ω ��F + ∇U ��2 + ��∇�V − ∇U ��2 dπ(x), 9 where the last equality is due to the orthogonality rela- tion (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Using a similar argument for deriving (31), the equivalence (31) itself, as well as the GP loss in (28), we get min LP-EPR(�V ) ⇐⇒ min LGP(�V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (32) Since �U minimizes the GP loss as is shown in the previous subsection, we conclude that �U minimizes the loss in (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Force projection loss In this subsection, we study the force projection (P- For) loss for approximating the projection of Fz onto the z-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Denote by �F (z) := � Σ Fz(y, z) dπ(y|z) (33) the projected force defined using the conditional distri- bution dπ(y|z) = pss(y, z)/�pss(z) dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (34) We can learn �F (z) via the following force projection loss LP-For( � G) = � Ω ��Fz(y, z) − � G(z) ��2 dπ(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (35) To justify (35), we note that � Ω ��Fz(y, z) − � G(z) ��2 dπ(y, z) = � Ω � |Fz(y, z)|2 + | � G(z)|2� dπ(y, z) − 2 � Ω Fz(y, z) · � G(z) dπ(y, z) =:P1 − 2P2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The term P2 can be simplified as P2 = � �Ω �� Σ Fz(y, z)dπ(y|z) � � G(z) �pss(z) dz = � �Ω �F (z) · � G(z) �pss(z) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Therefore, we have the equivalence min LP-For( � G) ⇐⇒ min �LP-For( � G), (36) where �LP-For( � G) := � �Ω �� �F (z) − � G(z) ��2�pss(z) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' From the analysis above we can conclude that �F(z) min- imizes the loss in (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' HJB equation for the reduced potential In this subsection, we show that the reduced potential �U satisfies the projected HJB equation �F · ∇z �U + |∇z �U|2 − D∆z �U − D∇z · �F = 0 , (37) with asymptotic BC �U → ∞ as |z| → ∞, or the reflecting BC ( �F + ∇z �U) · �n = 0 on ∂�Ω, where �n denotes the unit outer normal on ∂�Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We will only consider the rectangu- lar domain case here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The argument for the unbounded case is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Recall that pss(x) satisfies the FPE ∇ · (pssF ) − D∆pss = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (38) Integrating both sides of (38) on Σ with respect to y and utilizing the boundary condition Jss · n = 0, where Jss = pssF − D∇pss, we get ∇z · � � Σ Fzpss dy � − D∆z�pss = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (39) Taking (33) and (34) into account, we obtain ∇z · � �pss �F � − D∆z�pss = ∇z · � J = 0, (40) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=', a FPE for �pss(z) with the reduced force field �F , where � J := �pss �F −D∇z�pss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The corresponding boundary condition can be also derived by integrating the original BC Jss · n = 0 on Σ with respect to y for z ∈ ∂�Ω, which gives � J · �n = � �pss �F − D∇z�pss � �n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (41) Substituting the relation �pss(z) = exp � −�U(z)/D � into (40) and (41), we get (37) and the corresponding reflect- ing BC after some algebraic manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' STATE-DEPENDENT DIFFUSION COEFFICIENTS In this section, we study the EPR loss for NESS sys- tems with a state-dependent diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Consider the Ito SDEs dx(t) dt = F (x(t)) + √ 2Dσ(x(t)) ˙w (42) with the state-dependent diffusion matrix σ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Under the same assumptions as in the MT, we have the FPE ∇ · (pssF ) − D∇2 : (pssa) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (43) We show that the high dimensional landscape function U of (42) minimizes the EPR loss LV-EPR(V ) = � Ω |F v(x) + a(x)∇V (x)|2 a−1(x) dπ(x), (44) 10 where F v(x) := F (x) − D∇ · a(x) and |u|2 a−1(x) := u⊤a−1(x)u for u ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' To justify (44), we first note that (43) can be rewritten as ∇ · (pssF v − Da∇pss) = 0 , (45) which, together with the BC, implies the orthogonality relation � Ω � F v + a∇U � ∇W dπ = 0 (46) for a suitable test function W(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Following the same reasoning used in establishing (18) and utilizing (46), we have � Ω |F v + a∇V |2 a−1 dπ = � Ω ��F v + a∇U + a∇(V − U) ��2 a−1 dπ = � Ω |F v + a∇U ��2 a−1 dπ + � Ω ��a∇(V − U) ��2 a−1 dπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The last expression implies that U(x) is the unique min- imizer of LV-EPR(V ) up to a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The above derivation for the state-dependent diffusion case will permit us to construct the landscape for the chemical Langevin dynamics, which will be studied in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' PART 2: COMPUTATION Now we present the computational details and results omitted in the MT in the computation part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 2D MODELS AND COMPARISONS In this section, we will describe the computational setup and results for some 2D models which we utilize for the test of different formulations, including the toy model with known potential in the MT, a 2D biologi- cal system with a limit cycle [3] and a 2D multi-stable system [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We will also demonstrate the motivation for enhanced EPR and its advantage over other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Toy model and enhanced EPR In the toy model, we set the force field as F (x) = −(I + A) · ∇U0(x), (47) and choose the potential U0 = ((x − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5)2 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0))2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5(y − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5)2, (48) where x = (x, y)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We take the anti-symmetric matrix A = � 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 0 � , (49) which introduces a counter-clockwise rotation for a fo- cusing central force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' This sets up a simple non- equilibrium system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In this model, we have F (x) = −∇U0(x) + l(x), l(x) = −A · ∇U0(x) and l(x) · ∇U0(x) = 0 holds in the pointwise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' So, we have constructed a double-well non-reversible system with analytically known potential which can be used to verify the accu- racy of the learned potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We focus on the domain Ω = [0, 3] × [0, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Primarily, the single EPR loss works well for the toy model with a relatively large diffusion coefficient D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 1(A) in the MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' A slice plot of the poten- tial at y = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 3(A)) shows the EPR solution coin- cides well with the analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The relative root mean square error (rRMSE) and the relative mean abso- lute error (rMAE), which will be defined in Section F F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4, have mean and standard deviation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='099 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='010 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='081 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='013 over 3 runs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' However, when decreasing D to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='05, the samples from simulated invariant distribution mainly stay in the dou- ble wells and away from the transition region (orange 11 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' An illustration for the motivation of enhanced EPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (A) and (B) show the comparisons of the learned potentials and true solution on the line y = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 in the toy model with D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1 and D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='05, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (C) shows the filled contour plot of the potential learned by only the EPR loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The orange points are samples from the simulated invariant distribution with D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='05, While green points are enhanced samples simulated from a more diffusive distribution with D′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1, which are used in the enhanced EPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 3(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In this case, the double well do- main can still be learned well, yet the transition region, without enough samples, has not been effectively trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Thus, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 3(B), the single EPR result cap- tures the double wells, but cannot accurately connect them in the transition domain, which makes the left well a bit higher than the right one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The pointwise HJB loss with enhanced samples that better cover the transition domain thus helps the EPR loss with samples for small D, which mainly focuses on the local well domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Us- ing these enhanced samples for D′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1 (green points in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 3(A)), the enhanced EPR method performs much better in the transition domain between the two wells and thus agrees well with the true solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The above strategy is general, and we apply it to com- pute the landscape for all of the continued 2D problems and compare it with other methods in Section F F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 2D limit cycle model We apply our approach to the limit cycle dynamics with a Mexican-hat shape landscape [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Before proceeding to the concrete dynamical model, we have the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' For any SDEs like dx dt = F (x) + √ 2D ˙w, (50) the corresponding steady FPE is ∇ · (F pss) − D∆pss = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' If we make the transformation F → κF , D → κD in (50), then the steady state PDF pss(x) ∝ exp � −U(x) D � = exp � −κU(x) κD � is not changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The transformation only changes the timescale of the dynamics (50) from t0 to κt0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' However, this transformation changes the learned potential from U to κU if we utilize the drift κF (x) and noise strength κD in the system (50), which is helpful to set the scale of U to be O(1) by adjusting κ suitably for a specific problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' An alternative approach to accomplish this task is by choosing F to be κF in the EPR loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We take D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1 and consider the limit cycle dynamics dx dt = κ �α2 + x2 1 + x2 1 1 + y − ax � , (51) dy dt = κ τ0 � b − y 1 + cx2 � , (52) where the parameters are κ = 100, α = a = b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1, c = 100, and τ0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Here the choice of κ = 100 is to make U ∼ O(1) following [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We focus on the domain Ω = [0, 8] × [0, 8] and compute the potential landscape and force decomposition which is presented in the MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' As explained in the above paragraph, this corresponds to the case D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1/κ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='001 for the force field considered in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' B A C 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 EPR EPR True 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2 - True 一 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 Enhanced EPR 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='8 > y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='6 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 X + +12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Filled contour plots of the potential V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) for the toy model with D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='05 learned by (A) Enhanced EPR, (B) Naive HJB, and (C) Normalizing Flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The force field F (x) is decomposed into the gradient part −∇V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) (white arrows) and the non-gradient part (gray arrows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The length of an arrow denotes the scale of the vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The solid dots are samples from the simulated invariant distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 2D multi-stable model We also apply the enhanced approach to study the dynamics of a multi-stable system [5] dx dt = axn Sn + xn + bSn Sn + yn − k1x, (53) dy dt = ayn Sn + yn + bSn Sn + xn − k2y, (54) where the parameters are a = b = k1 = k2 = 1, S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5, and n = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We focus on the domain Ω = [0, 3] × [0, 3] and present the results for D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='01 in the MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Numerical comparisons In this subsection, we conduct a comparison study on the previous 2D problems to show the priority of our enhanced EPR approach over other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' For the toy model, we have the analytical solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' while for the other two 2D examples, we take the reference solution as the numerical solution of the steady FPE by a piecewise bilinear finite element method with fine rectangular grids and the least squares solver for the obtained sparse lin- ear system (a normalization condition � Ω pss(x)dx = 1 is added to fix the extra shifting degree of freedom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We use a fully connected neural network with 3 lay- ers and 20 hidden states as the potential V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We train the network with a batch size of 2048 and a learn- ing rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='001 by the Adam [35] optimizer for 3000 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We simulate the SDEs by the Euler-Maruyama scheme with reflecting boundaries on the boundary of the domain and obtain a dataset of size 10000 to approx- imate the invariant distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We update the dataset by one time step at each training iteration to make it closer to the invariant distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In the toy model, we try different scales to enhance samples and report the best performance (when D′ = 2D) for naive HJB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' For fairness, we use the same enhanced samples in enhanced EPR as naive HJB does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In SM, we denote the enhanced loss as λ1 LEPR +λ2 LHJB and use λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1, λ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 in the three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We can also use Gaussian disturbances of the SDE data to obtain enhanced data, as we do in the limit cycle problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We use D′ = 5D in the multi- stable problem for a better covering of the transition do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' For the comparison with normalizing flows, we train a neural spline flow [36] using the implementation from [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We repeat 4 blocks of the rational quadratic spline with 3 layers of 64 hidden units and a followed LU linear permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We train the flow model by Adam of the learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0001 for 20000 epochs, based on the same sample dataset as enhanced EPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We shift the potential to the origin by its minimum and focus on the domain D = {x ∈ Ω|V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) ≤ 20D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We then define the modified potential U m 0 (x) := min(U0(x), 20D), V m(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) := min(V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ), 20D) for the shifted potential U0(x) and V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) since only the potential values in the domain D is of practical interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We use the relative root mean square error (rRMSE) and the relative mean absolute error (rMAE) to describe the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' rRMSE = �� Ω |V m(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) − U m 0 (x)|2 dx � Ω |U m 0 (x)|2dx , (55) rMAE = � Ω |V m(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) − U m 0 (x)| dx � Ω |U m 0 (x)|dx .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' (56) We summarize the comparison of numerical errors for the 2D problems in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The advantages of enhanced EPR over both naive HJB and normalizing flow can be identified from the following points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' A B C 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2 1.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 X X X13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Slices of the learned 3D potential V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) in the Lorenz system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The solid dots are samples from the simulated invariant distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Comparisons on Numerical Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We report the mean and the standard deviation over 3 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Problem Method rRMSE rMAE Toy, D=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1 Enhanced EPR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='027±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='023±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='011 Naive HJB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='195±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='094±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='020 Normalizing Flow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='260±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='222±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='010 Toy, D=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='05 Enhanced EPR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='048±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='030±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='012 Naive HJB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='237±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='142±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='042 Normalizing Flow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='284±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='231±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='030 Limit Cycle Enhanced EPR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='052±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='029±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='016 Naive HJB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='107±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='048±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='019 Normalizing Flow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='255±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='210±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='015 Multi-stable Enhanced EPR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='040±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='022±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='005 Naive HJB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='103±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='053±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='006 Normalizing Flow 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='199±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='123±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='055 Without the guidance of EPR loss, naive HJB can not effectively optimize to the true solution with the heuristically chosen distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' As shown in Table I, the enhanced EPR significantly achieves much better performances than naive HJB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Also, in the toy model with D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='05, naively training by HJB leads to an unreliable solution in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 4(B) with relative errors larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Our computa- tional experiences show that the enhanced EPR is more robust than naive HJB and less sensitive to the enhanced data distribution and parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The enhanced EPR converges faster than the naive HJB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' For instance, in the toy model with D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1, the enhanced EPR has achieved rRMSE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='087 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='069 and rMAE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='066 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='013 in 2000 epochs, while the naive HJB can not attain the same level even after 3000 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Without information from the dynamics, the nor- malizing flow performs the worst only based on the simulated invariant distribution dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The learned potential tends to be rough and non- smooth at the edge of samples as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Thus the enhanced EPR explicitly utilizing the in- formation of the force field does help in more accu- rate training of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We further compare the potential landscape computed by different methods in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We remark that we omit the space {x|V (x) ≥ 30D} in both Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 3 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 4 since these domains are not of practical interest (their proba- bility is less than 10−9 according to the Gibbs form of the invariant distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The enhanced EPR presents the landscape more consistent with the simulated sam- ples and the true/reference solution than other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The decomposition of the force also shows better match- ing for the toy model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The normalizing flow captures the high probability domain but lacks information on the dy- namics, thus making its error larger than enhanced EPR and naive HJB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 3D MODELS In this section, we describe the computational setup for the Lorenz system in three dimensions and Ferrell’s three-ODE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We demonstrate the slices of the 3D 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 40 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 35 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 30 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 25 Z 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 15 -10 -5 12 1416 0 0 2 5 10 4 6 X 1514 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Streamlines of the projected force ˜ G(z) and filled contour plot of the reduced potential ˜V (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) for Ferrell’s three-ODE model learned by enhanced EPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' potential for the former and conduct the proposed di- mensionality reduction on the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 3D Lorenz system In this subsection, we apply our landscape construction approach to the 3D Lorenz system [38] with isotropic temporal Gaussian white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The Lorenz system has the form dx dt = β1(y − x), (57) dy dt = x (β2 − z) − y, (58) dz dt = xy − β3z, (59) where β1 = 10, β2 = 28 and β3 = 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We add the noise with strength D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' This model was also considered in [18] with D = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We obtain the enhanced data by adding Gaussian noises with standard deviation σ = 5 to the SDEs- simulation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We directly train the 3D potential V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) by enhanced EPR with λ1 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0, λ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 and present a slice view of the landscape in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The learned 3D potential agrees well with the simulated sam- ples and shows a butterfly-like shape as the original sys- tem does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Ferrell’s three-ODE model In this subsection, we consider Ferrell’s three-ODE model for a simplified cell cycle dynamics [29] denoted by x = [CDK1], y = [Plk1], z = [APC] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4 Plk1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='7 CDK115 for the concentration of CDK1, Plk1, and APC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We have the ODEs dx dt = α1 − β1x zn1 Kn1 1 + zn1 , (60) dy dt = α2 (1 − y) xn2 Kn2 2 + xn2 − β2y, (61) dz dt = α3 (1 − z) yn3 Kn3 3 + yn3 − β3z, (62) where α1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1, α2 = α3 = β1 = 3, β2 = β3 = 1, K1 = K2 = K3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5, n1 = n2 = 8, and n3 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We add the noise scale D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='01 with isotropic temporal Gaussian white noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' By taking the reduced variables z = (x, y)⊤, we can apply our force projection loss and enhanced loss to learn the projected force ˜G(x) and potential ˜V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ), and the results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We use three-layer net- works with 80 hidden states in this problem and en- hanced samples simulated from a more diffusive distribu- tion with D′ = 5D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We train the projected force ˜G(x) for 1000 epochs and then conduct enhanced EPR with λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1, λ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 for 4000 epochs to compute the pro- jected potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The obtained reduced potential shows a plateau in the centering region and a local-well tube domain along the reduced limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' HIGH DIMENSIONAL MODELS In this section, we apply our approach to 8D limit cy- cle dynamics [4] and 52D multistable dynamics [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We directly train the reduced force field ˜G(z) and poten- tial ˜V (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) according to the selected reduction variables suggested in the corresponding literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We use three- layer networks with 80 hidden states for both force and potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The training strategies are similar to previous examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 8D complex system We consider an 8D system in which the dynamics and parameters are the same as the supporting information of [4], and take CycB and Cyc20 as the reduction variable z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We set the mass in this problem as m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In [4], the noise strength D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0005 is not suitable for direct neural network training since the scale of the po- tential is O(10−5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Borrowing the idea in Section F F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2, we amplify the original force field F considered in [4] by κ = 1000 times, and take D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='01 for the trans- formed force field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' This amounts to set D = 10−5 for the original force field, which is even smaller than the case considered in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We simulate the SDEs without bound- aries first and then fix the dataset without updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We obtain the enhanced samples by adding Gaussian pertur- bations to the obtained dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Only the data within the FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Streamlines and limit sets of the projected force field of the 8D cell cycle model by two reduced variables CycB and Cdc20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The outer red circle is the stable limit cycle of the reduced force field corresponding to the yellow circle as the projection of the original high dimensional limit cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' The inner red circle, red dot and two green circles are stable and unstable limit sets of the reduced dynamics, which are virtual in high dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' biologically meaningful domain of [0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5]8 is utilized for computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We train the projected force ˜G(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) for 5000 epochs and conduct the enhanced EPR with λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1, λ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 for 10000 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Some essential features of the reduced potential and dynamics on the plane have been presented in MT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' In the SM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 7, we present a more thorough picture of the reduced dynamics for the 8D model than the MT Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' To be more specific, we further show two unstable 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='6 Cdc20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 CycB16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Projected force ˜ G(x) and potential ˜V (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) of the 52D double-well model learned by enhanced EPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' limit cycles of the projected force field, two green circles obtained by reverse time integration, in SM Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' They fall between the outer and inner stable limit cycles (inner and outer red circles), and the inner stable limit cycle and inner stable node (red dot in the center), which play the role of separatrices between the neighboring stable limit sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' This picture occurs as the result that the landscape of the considered system in the centering region is very flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' These inner limit sets are virtual in high dimensions, but they naturally appear in the reduced dynamics on the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Similar features might also occur in other reduced dynamics in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 52D multi-stable system We also apply our approach to a biological system with 52 ODEs constructed by [39] and take GATA6 and NANOG as the reduction variable z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We define Ai as the set of indices for activating xi and Ri as the set of indices for repressing xi, the corresponding relationships are defined as the 52D node network shown in [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' For i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=', 52, dxi dt = −kxi + � j∈Aj axn j Sn + xn j + � j∈Rj bSn Sn + xn j , (63) where a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='37, b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5, k = 1, S = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5, and n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We choose the noise strength D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We train the force ˜G(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' θ) for 500 epochs and conduct enhanced EPR with λ1 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0, λ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 for 500 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' We use enhanced samples simulated from a more diffusive distribution with D′ = 5D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 8, the projected force demonstrate the reduced dynamics and the depth of the constructed potential agrees well with the density of the sample points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='200 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='175 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='125 IANOG 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content='0 GATA617 [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dAzT4oBgHgl3EQf_P7B/content/2301.01946v1.pdf'} +page_content=' Waddington, The Strategy of the Genes (George 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+Liaoweiheng@gmail.com +Yang Song +BOSS Zhipin NLP Center +songyang@kanzhun.com +Tao Zhang +BOSS Zhipin +kylen.zhang@kanzhun.com +Dongyan Zhao +Peking University +zhaody@pku.edu.cn +Rui Yan‡ +Gaoling School of AI +Renmin University of China +ruiyan@ruc.edu.cn +ABSTRACT +Interview has been regarded as one of the most crucial step for +recruitment. To fully prepare for the interview with the recruiters, +job seekers usually practice with mock interviews between each +other. However, such a mock interview with peers is generally far +away from the real interview experience: the mock interviewers are +not guaranteed to be professional and are not likely to behave like +a real interviewer. Due to the rapid growth of online recruitment in +recent years, recruiters tend to have online interviews, which makes +it possible to collect real interview data from real interviewers. In +this paper, we propose a novel application named EZInterviewer, +which aims to learn from the online interview data and provides +mock interview services to the job seekers. The task is challenging +in two ways: (1) the interview data are now available but still of +low-resource; (2) to generate meaningful and relevant interview +dialogs requires thorough understanding of both resumes and job +descriptions. To address the low-resource challenge, EZInterviewer +is trained on a very small set of interview dialogs. The key idea is +to reduce the number of parameters that rely on interview dialogs +by disentangling the knowledge selector and dialog generator so +that most parameters can be trained with ungrounded dialogs as +well as the resume data that are not low-resource. Specifically, to +keep the dialog on track for professional interviews, we pre-train +a knowledge selector module to extract information from resume +in the job-resume matching. A dialog generator is also pre-trained +with ungrounded dialogs, learning to generate fluent responses. +* Both authors contributed equally to this research. +† Work done during an internship at BOSS Zhipin. +‡ Corresponding author: Rui Yan (ruiyan@ruc.edu.cn). +Permission to make digital or hard copies of all or part of this work for personal or +classroom use is granted without fee provided that copies are not made or distributed +for profit or commercial advantage and that copies bear this notice and the full citation +on the first page. Copyrights for components of this work owned by others than the +author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or +republish, to post on servers or to redistribute to lists, requires prior specific permission +and/or a fee. Request permissions from permissions@acm.org. +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +© 2023 Copyright held by the owner/author(s). Publication rights licensed to ACM. +ACM ISBN 978-1-4503-9407-9/23/02...$15.00 +https://doi.org/10.1145/3539597.3570476 +Then, a decoding manager is finetuned to combine information +from the two pre-trained modules to generate the interview ques- +tion. Evaluation results on a real-world job interview dialog dataset +indicate that we achieve promising results to generate mock in- +terviews. With the help of EZInterviewer, we hope to make mock +interview practice become easier for job seekers. +CCS CONCEPTS +• Computing methodologies → Natural language generation. +KEYWORDS +EZInterviewer, mock interview generation, knowledge-grounded +dialogs, online recruitment, low-resource deep learning +ACM Reference Format: +Mingzhe Li, Xiuying Chen, Weiheng Liao, Yang Song, Tao Zhang, Dongyan +Zhao, Rui Yan. 2023. EZInterviewer: To Improve Job Interview Performance +with Mock Interview Generator. In Proceedings of the Sixteenth ACM Inter- +national Conference on Web Search and Data Mining (WSDM ’23), February +27-March 3, 2023, Singapore, Singapore. ACM, New York, NY, USA, 9 pages. +https://doi.org/10.1145/3539597.3570476 +1 +INTRODUCTION +To make better preparations, job seekers practice mock interviews, +which aims to anticipate interview questions and prepare them +for what they might get asked in their real turn. However, the +outcome of such an approach is unsatisfactory, since those “mock +interviewers” do not have interview experience themselves, and +do not know what the real recruiters would be interested in. Mock +Interview Generation (MIG) represents a plausible solution to this +problem. Not only makes interviews more cost-effective, but mock +interview generators also appear to be feasible, since much can be +learned about the job seekers from their resumes, as can the job +itself from the job description (JD). An illustration of MIG task is +shown in Figure 1. +There are two main challenges in this task. One is that the +knowledge-grounded interviews are extremely time-consuming +and costly to collect. Without a sufficient amount of training data, +arXiv:2301.00972v1 [cs.CL] 3 Jan 2023 + +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +Mingzhe Li et al. +Figure 1: An example of the Mock Interview Generation task. +Based on the candidate’s work experience and the current di- +alog on the experience of web page development, the system +generates an interview question “If a product needs a three- +level classification selection, which component would you +use and how to achieve it?”. +the performance of such dialog generation models drops dramati- +cally [37]. The second challenge is to make the knowledge-grounded +dialog relevant to the candidate resume, job description, and previ- +ous dialog utterances. This makes MIG a complex task involving +text understanding, knowledge selection, and dialog generation. +In this paper, we propose EZInterviewer, a novel mock interview +generator, with the aim of making interviews easier to prepare. The +key idea is to train EZInterviewer in a low-resource setting: the +model is first pre-trained on large-scale ungrounded dialogs and +resume data, and then fine-tuned on a very small set of resume- +grounded interview dialogs. Specifically, the knowledge selector +consists of a resume encoder to encode the resume, and a key-value +memory network with mask self-attention mechanism, responsible +for selecting relevant information in the resume to focus on to help +generate the next interview utterance. The dialog generator also +has two components, a context encoder which encodes the current +dialog context, and a response decoder, responsible for generating +the next dialog utterance without knowledge from the resumes. +This knowledge-insensitive dialog generator is coordinated with +the knowledge selector by a decoding manager that dynamically +determines which component is activated for utterance generation. +It is noted that the number of parameters in the decoding man- +ager can be small, therefore it only requires a small number of +resume-grounded interview dialogs. Extensive experiments on real- +world interview dataset demonstrate the effectiveness of our model. +To summarize, our contributions are three-fold: +• We introduce a novel Mock Interview Generation task, which +is a pilot study of intelligent online recruitment with potential +commercial values. +• To address the low-resource challenge, we propose to reduce +the number of parameters that rely on interview dialogs by dis- +entangling knowledge selector and dialog generator so that the +majority of parameters can be trained with large-scale ungrounded +dialog and resume data. +• We propose a novel model to jointly process dialog contexts, +candidate resumes, and job descriptions and generate highly rele- +vant, knowledge-aware interview dialogs. +2 +RELATED WORK +Multi-turn response generation aims to generate a response that is +natural and relevant to the entire context, based on utterances in its +previous turns. [36] concatenated multiple utterances into one sen- +tence and utilized RNN encoder or Transformer to encode the long +sequence, simplifying multi-turn dialog into a single-turn dialog. +To better model the relationship between multi-turn utterances, +[4, 10] introduced interaction between utterances after encoding +each utterance. +As human conversations are almost always grounded with exter- +nal knowledge, the absence of knowledge grounding has become +one of the major gaps between current open-domain dialog systems +and real human conversations [8, 24, 35]. A series of work [20, 29] +focused on generating a response based on the interaction between +context and unstructured document knowledge, while a few oth- +ers [22, 33] introduced knowledge graphs into conversations. These +models, however, usually under-perform in a low-resource setting. +To address the low resource problem, [16] proposed to enhance +the context-dependent cross-lingual mapping upon the pre-trained +monolingual BERT representations. [28] extended the meta-learning +algorithm, which utilized knowledge learned from high-resource +domains to boost the performance of low-resource unsupervised +neural machine translation. Different from the above methods, [37] +proposed a disentangled response decoder in order to isolate pa- +rameters that depend on knowledge-grounded dialogs from the +entire generation model. Our model takes a step further, taking +into account the changes in attention on knowledge in multi-turn +dialog scenarios. +3 +MODEL +3.1 +Problem Formulation +For an input multi-turn dialog context 𝑈 = {𝑢1,𝑢2, . . . ,𝑢𝑚} be- +tween a job candidate and an interviewer, where 𝑢𝑖 represents the +𝑖-th utterance, we assume there is a ground truth textual interview +question 𝑌 = {𝑦1,𝑦2, . . . ,𝑦𝑛}. 𝑚 is the utterance number in the +dialog context and 𝑛 is the total number of words in question 𝑌. +In the 𝑖-th utterance, 𝑢𝑖 = {𝑥𝑖 +1,𝑥𝑖 +2, . . . ,𝑥𝑖 +𝑇 𝑖𝑢 }. Meanwhile, there is a +candidate resume 𝑅 = {(𝑘1, 𝑣1), (𝑘2, 𝑣2), . . . , (𝑘𝑇𝑟 , 𝑣𝑇𝑟 )} correspond- +ing to the candidate in the interview, which has 𝑇𝑟 key-value pairs, +and each of which represents an attribute in the resume. For the +job-resume matching pre-training task, there is an external job +description 𝐽 = {𝑗1, 𝑗2, . . . , 𝑗𝑇𝑗 }, which has 𝑇𝑗 words. The goal is to +generate an interview question 𝑌 +′ that is not only coherent with the +dialog context 𝑈 but also pertinent to the job candidate’s resume 𝑅. +3.2 +System Overview +In this section, we propose our Low-resource Mock Interview Gen- +erator (EZInterviewer) model, which is divided into three parts as +shown in Figure 2: + +O +100%10:38 +100%10:47 +99%11:09 +MyOnlineResume +Preview +< +< +Mock Interview +Web Front-end +Expected Position +Hello, I'm an undergraduate, and I +Development Engineer +am confident that I am qualified for +Python 8-9K + Fresh graduate + Undergraduate +the intern position of web front-end +San Francisco +engineer.I hope you can see my +information. +HR +Work Experience +Do you have experience in +Job Description +Company +Emini programs or pc website +2019.01-now> +Web front-end development +Web front-end development +Webpack +development? +Content. +GIT +Gulp +JavaScript +Vue +:Iusedtodevelopfront-endwebpagesbasedor +Adjust the style of the system +:HTML and CSS. +I'm sorry that I have not done any +back-stage on the PC front-end and call +mini program work, but I used to +Web front-end +Vue +Mini program +develop Web page in the past. +the interface to decelop a small part of +thefunction +Work closely with the back-end devel- +Project Experience +:Let's do a test. If a product needs a +opment team to ensure limited code +:three-level .classification. selection... +which component would you use +:docking,optimize front-end perfor- +and how to achieve it? +Education Experience +④ +:mance, and participate in mobile inter- +:face development and architecture +university +2019-2022 +:design; +Message +? +Undergraduate +Computer ScienceEZInterviewer: To Improve Job Interview Performance with Mock Interview Generator +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +Job Desc +Job Encoder +Resume +Resume Encoder +Multi-turn +Interview History +Self Attention + Add & Norm +Position-Wise FeedForward +[CLS] +[CLS] +[CLS] +... +... +... +Utterance +States +Masked Self Attention +Cross Attention Manager +Add & Norm +Position-Wise FeedForward +Decoder Input +xN +Linear +Linear +Linear +Decoder +State +Utterance +States +Cross +Attention +Cross +Attention +xN +updated output +Fusion gate +Pretrained +Soft Target +Updated +Distribution +One-hot +Target +Masked +Self-attention +Visible Matrix +Cross +Attention +Transfer +Memory +Job-resume +Matching +Knowledge Selector (Pretraining) +Dialog Generator (Pretraining) +Decoding Manager +Self Attention +Key-Value +Memory Network +Figure 2: Overview of EZInterviewer, which consists of three parts: (1) Knowledge Selector selects salient knowledge infor- +mation from the candidate resume; (2) Dialog Generator predicts the next word without knowledge of resumes; (3) Decoding +Manager coordinates the output from knowledge selector and dialog generator to produce the interview question. +• Dialog Generator predicts the next word of a response based on +the prior sub-sequence. In our model, we pre-train it by large-scale +ungrounded dialogs. +• Knowledge Selector selects salient knowledge information from +the candidate resume for interview question generation. In our +model, we augment the ability of the knowledge selector by em- +ploying it to perform job-resume matching. +• Decoding Manager coordinates the output from knowledge +selector and dialog generator to predict the interview question. +It is important to note that to train an EZInterviewer model, +two pre-train techniques are employed. Firstly, we pre-train the +knowledge selector in a job-matching task. This is because while +it is hard to attend to appropriate content in a resume just on its +own, the salient information in a resume can be identified in a +job-resume matching task [13, 34]. Secondly, the context encoder +and response decoder of the dialog generator are pre-trained with +a large scale of ungrounded dialogs, so as to predict the next word +of response based on the prior sub-sequence. Finally, the decoding +manager, which relies on a few parameters, coordinates the two +components to generate knowledge grounded interview utterance. +3.3 +Dialog Generator +Context Encoder. Instead of processing the dialog context as a +flat sequence, we employ a hierarchical encoder [3] to capture intra- +and inter-utterance relations, which is composed of a local sentence +encoder and a global context encoder. For the sentence encoder, to +model the semantic meaning of the dialog context, we learn the +representation of each utterance 𝑢𝑖 by a self-attention mechanism +(SAM) initialized by BERT [5]: +ℎ𝑖 +𝑗 = SAMu(𝑒(𝑥𝑖 +𝑗),ℎ𝑖 +∗). +(1) +We extract the state at “[cls]” position to denote the utterance state, +abbreviated as ℎ𝑖. Apart from the local information exchange in +each utterance, we let information flow across multi-turn context: +ℎ𝑐 +𝑡 = SAMc(ℎ𝑡,ℎ𝑐 +∗), +(2) +where ℎ𝑐 +𝑡 denotes the hidden state of the 𝑡-th utterance in SAMc. +Response Decoder. Response decoder is responsible for under- +standing the previous dialog context and generates the response +without the knowledge of resume information [19]. Our decoder +also follows the style of Transformer. +Concretely, we first apply the self-attention on the masked de- +coder input, obtaining𝑑𝑡. Based on𝑑𝑡 we compute the cross-attention +scores over previous utterances: +𝛼𝑐 +𝑡 = ReLU([𝑑𝑡𝑊𝑑 (ℎ𝑐 +𝑖𝑊ℎ)𝑇 ]). +(3) +The attention weights 𝛼𝑐 +𝑡 is then used to obtain the context vectors +as𝑐𝑡 = �𝑚 +𝑖=1 𝛼𝑐 +𝑡 ℎ𝑐 +𝑖 . The context vectors𝑐𝑡, treated as salient contents +of various sources, are concatenated with the decoder hidden state +𝑑𝑡 to produce the distribution over the target vocabulary: +𝑃𝑤 +𝑣 = Softmax (𝑊𝑜 [𝑑𝑡;𝑐𝑡]) . +(4) +Pre-training process. While interview dialogs are hard to come +by, online conversation is abundant on the internet, and can be +easily collected. Hence, we pre-train the dialog generator on un- +grounded conversations. Concretely, during pre-training process, +we employ the context encoder to first encode the multi-turn pre- +vious dialog context. Then, at the 𝑡-th decoding step, we use the +response decoder to predict the 𝑡-th word in the response. We set +the loss as the negative log likelihood of the target word 𝑦𝑡: +𝐿𝑜𝑠𝑠𝑔 = − 1 +𝑛 +�𝑛 +𝑡=1 log 𝑃𝑤 +𝑣 (𝑦𝑡). +(5) +3.4 +Knowledge Selector +Resume Encoder. As shown in Figure 2, a resume contains several +key-value pairs (𝑘𝑖, 𝑣𝑖). Most of key and value fields include a single +word or a phrase such as “skills” or “gender”, and we can obtain the +feature representation through an embedding matrix. Concretely, +for each key or value field with a single word or a phrase, we estab- +lish a corresponding resume embedding matrix 𝑒𝑖𝑟 that is different +from the previous one. Then we use the resume embedding matrix +to map each field word 𝑘𝑖 or 𝑣𝑖 into to a high-dimensional vector +space, denoted as 𝑒𝑖𝑟 (𝑘𝑖) or 𝑒𝑖𝑟 (𝑣𝑖). For fields with more than one +word such as “work experience” or “I used to...”, we denote them as +𝑣𝑖 = (𝑣1 +𝑖 , ...𝑣𝑙𝑖 +𝑖 ), where 𝑙𝑖 denotes the word number of the current + +Birthday +19980501 +Gender +MaleWSDM ’23, February 27-March 3, 2023, Singapore, Singapore +Mingzhe Li et al. +field. We first process them through the previous word embedding +matrix 𝑒, then there is an SAMR, similar with SAMu in Section 3.3, +to model the temporal interactions between words: +ℎ𝑟𝑖 +𝑡 = SAMR(𝑒(𝑣 𝑗 +𝑖 ),ℎ𝑟𝑖 +𝑡−1). +(6) +We use the last hidden state of the SAMR, i.e., ℎ𝑟𝑖 +𝑙𝑖 to denote the +overall representation for field 𝑣𝑖. +For brevity, in the following sections, we use ℎ𝑘 +𝑖 and ℎ𝑣 +𝑖 to denote +the encoded key-value pair (𝑘𝑖, 𝑣𝑖) in the resume. +Masked Self-attention. Traditional self-attention can be used +to update representation of each resume item due to its flexibility in +relating two elements in a distance-agnostic manner [17]. However, +as shown in [21], too much knowledge incorporation may divert the +representation from its correct meaning, which is called knowledge +noise (KN) issue. In our scenario, the information in the resume +is divided into several parts, i.e., basic personal information, work +experiences and extended work, each of which contains variable +number of items. The items within each part are closely connected, +while different parts can be considered as different domains, and +the interaction may introduce a certain amount of noise. To over- +come this problem, we introduce a visible matrix, in which items +belonging to the same part are visible to each other, while the visi- +bility degree between items is determined by the cosine similarity +of semantic representations, i.e., 𝐶𝑖,𝑗 = cos_sim(ℎ𝑣 +𝑖 ,ℎ𝑣 +𝑗 ). Then, the +scaled dot-product masked self-attention is defined as: +𝛼𝑖,𝑗 = +exp +� +(ℎ𝑘 +𝑖 𝑊𝑞)𝐶𝑖,𝑗 (ℎ𝑘 +𝑗𝑊𝑘)𝑇 � +�𝑇𝑟 +𝑛=1 exp +� +(ℎ𝑘 +𝑖 𝑊𝑞)𝐶𝑖,𝑛(ℎ𝑘 +𝑗𝑊𝑘)𝑇 +� , +(7) +ˆℎ𝑣 +𝑖 = +∑︁𝑇𝑟 +𝑗=1 +𝛼𝑖,𝑗ℎ𝑣 +𝑗 +√ +𝑑 +, +(8) +where 𝑑 stands for hidden dimension and 𝐶 is the visible matrix. +ˆℎ𝑣 +𝑖 is then utilized as the updated resume value representation. +Key-Value Memory Network. The goal of key matching is to +calculate the relevance between each attribute of the resume and +the previous dialog context. Given dialog context ℎ𝑖, for the 𝑗-th +attribute pair (𝑘𝑗, 𝑣𝑗), we calculate the probability of ℎ𝑖 over 𝑘𝑗, +i.e., 𝑃(𝑘𝑗 |ℎ𝑖), as the matching score 𝛽𝑖,𝑗. To this end, we exploit the +context representation ℎ𝑖 to calculate the matching score: +𝛽𝑖,𝑗 = +exp +� +ℎ𝑖𝑊𝑎ℎ𝑘 +𝑗 +� +�𝑇𝑟 +𝑛=1 exp +� +ℎ𝑖𝑊𝑎ℎ𝑘𝑛 +� . +(9) +Since context representation ℎ𝑖 and resume key representation ℎ𝑘 +𝑗 +are not in the same semantic space, we use a trainable key matching +parameter𝑊𝑎 to transform these representations into a same space. +As the relevance between context ℎ𝑖 and each pair in the resume +table (𝑘𝑗, 𝑣𝑗), the matching score 𝛽𝑖,𝑗 can help to capture the most +relevant pair for generating a correct question. Therefore, as shown +in Equation 10, the knowledge selector reads the information 𝑀𝑖 +from KVMN via summing over the stored values, and guides the +follow-up response generation, so we have: +𝑀𝑖 = +∑︁𝑇𝑟 +𝑗=1 𝛽𝑖,𝑗 ˆℎ𝑣 +𝑗, +(10) +where ˆℎ𝑣 +𝑗 is the representation of value 𝑣𝑗, and 𝛽𝑖,𝑗 is the matching +score between dialog context ℎ𝑖 and key 𝑘𝑗. +Pre-training Process. In practice, the resume knowledge con- +tains a variety of professional and advanced scientific concepts such +as “Web front-end”, “HTML”, and “CSS”. These technical terms are +difficult to understand for people not familiar with the specific +domain, not to mention for the model that is not able to access +a large-scale resume-grounded dialog dataset. Hence, it would be +difficult for the knowledge selector to understand the resume con- +tent and previous context about the resume, so as to select the next +resume pair to focus on. +On the other hand, we notice that in job-resume matching task, +it is crucial to capture the decisive information in the resume to +perform a good matching. For example, recruiters may tend to hire +the candidate with particular experiences among several candidates +with similar backgrounds [34]. Intuitively, the key-value pair that is +important for job-resume matching is also the key factor to consider +in a job interview. Hence, if we can let the model learn the salient +information in the resume by performing the job-resume matching +task on large-scale job-resume data, then it would also bring benefits +for selecting salient information in interview question generation. +Concretely, we use the job description to attend to the resume to +perform a job-resume matching task, as a pre-training process for +knowledge selector module. As shown in Figure 2, the Job Encoder +encodes the job description by a SAMjd: +ℎ𝑗𝑑 +𝑖 += SAMjd(𝑒(𝑗𝑖),ℎ𝑗𝑑 +𝑖−1), +(11) +where 𝑗𝑖 denotes the 𝑖-th word in the job description, and 𝑒(𝑗𝑖) +is mapped by the previous embedding matrix 𝑒. We use the final +hidden state of the SAMjd, i.e., ℎ𝑗𝑑 +𝑇𝑗 as the overall representation +for the description, abbreviated as ℎ𝑗𝑑. ℎ𝑗𝑑 plays a similar part as +the context representation ℎ𝑖, which first attends to the keys in the +resume, and then is used to “weightedly” read the values in the +resume. We use 𝑚𝑗𝑑 to denote the weighted read result. +In the training process, we first pre-train the knowledge selector +by job-resume matching task, which can be formulated as a classi- +fication problem [26]. The objective is to maximize the scores of +positive samples while minimizing that of the negative samples. +Specifically, we concatenateℎ𝑗𝑑 and𝑚𝑗𝑑 since vector concatenation +for matching is known to be effective [27]. Then the concatenated +vector is fed to a multi-layer, fully-connected, feed-forward neural +network, and the job-resume matching score 𝑠𝑗𝑟 is obtained as: +𝑠𝑗𝑟 = 𝜎 +� +𝐹𝑠 ([ℎ𝑗𝑑;𝑚𝑗𝑑]]) +� +, +(12) +where [; ] denotes concatenation operation, and the outputs are the +probabilities of successfully matching. We use the job-resume pairs +in interviews as positive samples, and then use the job-resume pairs +without interviews as negative instances. +After pre-training, the job description is replaced by the context +representations, while the key matching and value combination +processes remain the same. We use a knowledge memory 𝑀 to store +the selection result, where each slot stores the value combination +result 𝑀𝑖 in Equation 10. + +EZInterviewer: To Improve Job Interview Performance with Mock Interview Generator +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +3.5 +Decoding Manager +The decoding manager is supposed to generate the proper word +based on the knowledge memory and the response decoder. Our +idea is inspired by an observation on the nature of interview dialogs: +despite the fact that a dialog is based on the resume, words and utter- +ances in the dialog are not always related to resume. Therefore, we +postulate that formation of a response can be decomposed into two +uncorrelated actions: (1) selecting a word according to the context +to make the dialog coherent (corresponding to the dialog generator); +(2) selecting a word according to the extra knowledge memory to +ground the dialog (corresponding to the knowledge selector). The +two actions can be independently performed, which becomes the +key reason why the large resume-job matching and ungrounded +dialog datasets, although seemingly unrelated to interview dialogs, +can be very useful in an MIG task. +Note that in Section §3.4, we store the selected knowledge 𝑀𝑖 in +a knowledge memory 𝑀. To select a word based on it, similar to +the response decoder, we use 𝑑𝑡 to attend to each slot of knowledge +memory, and we can obtain the knowledge context vector 𝑔𝑘 +𝑡 and +the output decoder state 𝑑𝑘𝑜 +𝑡 . +The response decoder and knowledge selector are controlled by +the decoding manager with a “fusion gate” to decide how much +information from each side should be focused on at each step of +interview question prediction. +𝛾𝑓 = 𝜎 (𝐹𝑚(𝑑𝑡)) , +(13) +where 𝑑𝑡 is the 𝑡-th decoder hidden state. Then, the probability to +predict word 𝑦𝑡 can be formulated as: +𝑑𝑜 +𝑡 = 𝛾𝑓 𝑑𝑤𝑜 +𝑡 ++ (1 − 𝛾𝑓 )𝑑𝑘𝑜 +𝑡 , +(14) +𝑃𝑣 = softmax �𝑊𝑣𝑑𝑜 +𝑡 + 𝑏𝑣 +� . +(15) +As for the optimization goal, generation models that use one- +hot distribution optimization target always suffer from the over- +confidence issue, which leads to poor generation diversity [32]. +Hence, aside from the ground truth one-hot label 𝑃, we also propose +a soft target label 𝑃𝑤 +𝑣 (see Equation 4), which is borrowed from the +pre-trained Dialog Generator in Section 3.3. Forcing the decoding +manager to simulate the pre-trained decoder can help it learn the +context of the interview dialog. We combine the one-hot label with +the soft label by an editing gate 𝜆, as shown in Figure 2. Concretely, +a smooth target distribution 𝑃 ′ is proposed to replace the hard +target distribution 𝑃 as: +𝑃 ′ = 𝜆𝑃 + (1 − 𝜆)𝑃𝑤 +𝑣 . +(16) +where 𝜆 ∈ [0, 1] is an adaption factor, 𝑃𝑤 +𝑣 is obtained from Equa- +tion 4, and 𝑃 is the hard target as one-hot distribution which assigns +a probability of 1 for the target word 𝑦𝑡 and 0 otherwise. +4 +EXPERIMENTAL SETUP +4.1 +Dataset +In this paper, we conduct experiments on a real-world dataset pro- +vided by “Boss Zhipin” 1, the largest online recruiting platform +in China. To protect the privacy of candidates, user records are +anonymized with all personal identity information removed. The +1https://www.zhipin.com +Table 1: Statistics of the datasets used in the experiments. +Statistics +Values +Interview Dialog Dataset +Total number of resumes +12,666 +Total number of dialog utterances +49,214 +Avg turns # per dialog context +4.47 +Avg words # per utterance +13.18 +Job-Resume Dataset +Key-value pairs # per resume +22 +Avg words # per work experience in resume +72.80 +Avg words # per self description in resume +51.13 +Avg words # per job description +74.26 +Ungrounded Dialog Dataset +Total number of context-response pairs +2,995,000 +Avg turns # per dialog context +4 +Avg words # per utterance +15.15 +dataset includes 12,666 resumes, 8,032 job descriptions, and 49,214 +interview dialog utterances. The statistics of the dataset is summa- +rized in Table 1. We then tokenize each sentence into words with +the benchmark Chinese tokenizer toolkit “JieBa” 2. +To pre-train the knowledge selector module, we use a job-resume +matching dataset [34], again from “Boss Zhipin”. The training +set and the validation set include 355,000 and 1,006 job-resume +pairs, respectively. To pre-train dialog generator, we choose Weibo +dataset [2], which includes a massive number of multi-turn con- +versations collected from “Weibo”3. The data includes 2,990,000 +context-response pairs for training and 5,000 pairs for validation. +The details are also summarized in Table 1. +4.2 +Comparisons +We compare our proposed model against traditional knowledge- +insensitive dialog generation baselines, and knowledge-aware dia- +log generation baselines. +• Knowledge-insensitive dialog generation baselines: +Transformer [30]: is based solely on attention mechanisms. +BERT [5]: initializes Transformer with BERT as the encoder. Di- +aloGPT [36]: proposes a large, tunable neural conversational re- +sponse generation model trained on more conversation-like ex- +changes. T5-CLAPS [14]: generates samples for contrastive learn- +ing by adding small and large perturbations, respectively. +• Knowledge-aware dialog generation baselines: +TMN [6]: is built upon a transformer architecture with an ex- +ternal memory hosting the knowledge. ITDD [20]: incrementally +encodes multi-turn dialogs and knowledge and decodes responses +with a deliberation technique. DiffKS [38]: utilizes the differential +information between selected knowledge in multi-turn conversa- +tion for knowledge selection. DRD [37]: tackles the low-resource +challenge with pre-training techniques using ungrounded dialogs +and documents. DDMN [31]: dynamically keeps track of dialog +context for multi-turn interactions and incorporates KB knowledge +2https://github.com/fxsjy/jieba +3https://www.weibo.com + +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +Mingzhe Li et al. +Table 2: Comparing model performance on full dataset: automatic evaluation metrics. +BLEU-1 +BLEU-2 +BLEU-3 +BLEU-4 +Extrema +Average +Greedy +Dist-1 +Dist-2 +Entity F1 +Cor +Knowledge-insensitive dialog generation +Transformer [30] +0.5339 +0.3811 +0.2836 +0.2530 +0.4859 +0.7673 +0.6803 +0.0928 +0.3157 +0.3606 +0.2711 +BERT [5] +0.5671 +0.3864 +0.2735 +0.2583 +0.4861 +0.7669 +0.6792 +0.0947 +0.3558 +0.3711 +0.2894 +DialoGPT [36] +0.5722 +0.4015 +0.3004 +0.2697 +0.4858 +0.7670 +0.6814 +0.1001 +0.3620 +0.3843 +0.3002 +T5-CLAPS [14] +0.5846 +0.4126 +0.3020 +0.2783 +0.4837 +0.7851 +0.6674 +0.0970 +0.3702 +0.3549 +0.2870 +Knowledge-aware dialog generation +TMN [6] +0.5437 +0.3891 +0.2963 +0.2630 +0.4841 +0.7655 +0.6811 +0.0996 +0.3299 +0.3830 +0.2652 +ITDD [20] +0.5484 +0.4009 +0.2929 +0.2656 +0.4833 +0.7650 +0.6859 +0.1055 +0.3703 +0.3661 +0.2715 +DiffKS [38] +0.5617 +0.3898 +0.2776 +0.2441 +0.4826 +0.7830 +0.6752 +0.0937 +0.3612 +0.3672 +0.2750 +DRD [37] +0.5711 +0.4001 +0.2914 +0.2548 +0.4824 +0.7813 +0.6783 +0.0867 +0.3661 +0.3825 +0.2883 +DDMN [31] +0.5693 +0.4065 +0.2968 +0.2694 +0.4831 +0.7655 +0.6811 +0.0944 +0.3640 +0.3754 +0.2869 +Persona [9] +0.5532 +0.3829 +0.2715 +0.2377 +0.4823 +0.7822 +0.6783 +0.0911 +0.3598 +0.3833 +0.2928 +EZInterviewer +0.6106 +0.4320 +0.3284 +0.2917 +0.4893 +0.7884 +0.6886 +0.1071 +0.3747 +0.3927 +0.3145 +No Pre-train +0.5738 +0.4029 +0.2929 +0.2599 +0.4846 +0.7833 +0.6831 +0.0981 +0.3673 +0.3819 +0.3007 +w/o KM +0.5795 +0.4127 +0.3069 +0.2754 +0.4847 +0.7841 +0.6762 +0.0979 +0.3685 +0.3803 +0.3010 +w/o KS +0.5775 +0.4122 +0.3067 +0.2746 +0.4781 +0.7668 +0.6787 +0.1003 +0.3691 +0.3848 +0.2994 +w/o LS +0.6007 +0.4232 +0.3176 +0.2821 +0.4869 +0.7863 +0.6832 +0.0969 +0.3664 +0.3902 +0.3127 +into generation. Persona [9]: introduces personal memory into +knowledge selection to address the personalization issue. +4.3 +Implementation Details +We implement our experiments in TensorFlow [1] on an NVIDIA +GTX 1080 Ti GPU. For our model and all baselines, we follow the +same setting as described below. We truncate input dialog to 100 +words with 20 words in each utterance, as we did not find significant +improvement when increasing input length from 100 to 200 tokens. +The minimum decoding step is 10, and the maximum step is 20. +The word embedding dimension is set to 128 and the number of +hidden units is 256. Experiments are performed with a batch size +of 256, and the vocabulary is comprised of the most frequent 50k +words. We use Adam optimizer [12] as our optimizing algorithm. +We selected the 5 best checkpoints based on performance on the +validation set and report averaged results on the test set. Note that +for better performance, our model is built based on BERT, and the +decoding process is the same as Transformer [30]. Finally, due to +the limitation of time and memory, small settings are used in the +pre-trained baselines. +4.4 +Evaluation Metrics +To evaluate the performance of EZInterviewer against baselines, +we adopt the following metrics widely used in existing studies. +Overlap-based Metric. Following [18], we utilize BLEU score +[25] to measure n-grams overlaps between ground-truth and gener- +ated response. In addition, we apply Correlation (Cor) to calculate +the words overlap between generated question and job description, +which measures how well the generated questions line up with the +recruitment intention. +Embedding Metrics. We compute the similarity between the +bag-of-words (BOW) embeddings of generated results and reference +to capture their semantic matching degrees [11]. In particular we +adopt three metrics: 1) Greedy, i.e., greedily matching words in two +Table 3: Human evaluation results on: Readability (Read), +Informativeness (Info), Meaningfulness (Mean), Usefulness +(Use), Relevance (Rel), and Coherence (Coh). +Model +Dialog-level +Interview-level +Read +Info +Mean +Use +Rel +Coh +DiffKS +1.79 +2.01 +1.87 +2.03 +1.99 +2.10 +DDMN +1.97 +1.83 +1.63 +2.12 +2.14 +1.91 +DRD +2.05 +2.11 +2.09 +2.08 +2.17 +2.02 +EZInterviewer +2.42▲ +2.51▲ +2.39▲ +2.46▲ +2.57▲ +2.38▲ +utterances based on cosine similarities; 2) Average, cosine similarity +between the averaged word embeddings in two utterances [23]; +3) Extrema, cosine similarity between the largest extreme values +among the word embeddings in the two utterances [7]. +Distinctness. The distinctness score [15] measures word-level +diversity by calculating the ratio of distinct uni-gram and bi-grams +in generated responses. +Entity F1. Entity F1 is computed by micro-averaging precision +and recall over knowledge-based entities in the entire set of sys- +tem responses, and evaluates the ability of a model to generate +relevant entities to achieve specific tasks from the provided knowl- +edge base [31]. The entities we use are extracted from an entity +vocabulary provided by “Boss Zhipin”. +Human Evaluation Metrics. We further employ human eval- +uations aside from automatic evaluations. Three well-educated +annotators from different majors are hired to evaluate the quality +of generated responses, where the evaluation is conducted in a +double-blind fashion. In total 100 randomly sampled responses gen- +erated by each model are rated by each annotator on both dialog +level and interview level. We adopt the Readability (is the response +grammatically correct?) and Informativeness (does the response +include informative words?) to judge the quality of the generated + +EZInterviewer: To Improve Job Interview Performance with Mock Interview Generator +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +responses on the dialog level. On the interview level, we adopt +Meaningfulness (is the generated question meaningful?), Usefulness +(is the question worth the job candidate preparing in advance?), +Relevance (is the question relevant to the resume?) and Coherence (is +the generated text coherent with the context?) to assess the overall +performance of a model and the quality of user experience. Each +metric is given a score between 1 and 3 (1 = bad, 2 = average, 3 = +good). +5 +EXPERIMENTAL RESULT +5.1 +Overall Performance +Automatic evaluation. The comparison between EZInterviewer +and state-of-the-art generative baselines is listed in Table 2. +We take note that the knowledge-aware dialog generation mod- +els outperform traditional dialog models, suggesting that utilizing +external knowledge introduces advantages in generating relevant +response. We also notice the pre-train based model DRD outper- +forms other baselines, showing that initializing parameters by pre- +training on large-scale data can lead to a substantial improvement +in performance. It is worth noting some models achieve better En- +tity F1 but a lower BLEU score; this suggests that those models tend +to copy necessary entity words from the knowledge but are not +able to use them properly. +EZInterviewer outperforms baselines on all automatic metrics. +Firstly, our model improves BLEU-1 by 6.92% over DRD. On the Dis- +tinctness metric Dist-1, our model outperforms DialoGPT by 6.99%, +suggesting that the generated interview questions are diversified +and personalized with different candidates’ resumes. Moreover our +model attains a good score of 0.3927 on entity F1, which evaluates +the degree to which the generated question is grounded on the +knowledge base. Finally, Cor score of 0.3145 suggests the ques- +tions generated by EZInterviewer is in line with the job description, +hence reflect the intention of the recruiters. Overall the metrics +demonstrate that our model successfully learns an interviewer’s +points of interest in a resume, and incorporates this knowledge into +interview questions properly. +Human evaluation. The results of human evaluations on all +models are listed in Table 3. EZInterviewer is the top performer on +all the metrics. Specifically, our model outperforms DiffKS by 35.20% +on Readability, suggesting that EZInterviewer manages to reduce +the grammatical errors and improve the readability of the generated +response. As for the Informativeness metric, our model scores 0.68 +higher than DDMN. This indicates that EZInterviewer captures +salient information in the resume. On the interview level, EZInter- +viewer’s Usefulness score is 18.27% better than DRD, demonstrating +its capabilities to help job seekers to pick the right questions to +prepare. On Relevance metric, our model outperforms all baselines +by a considerable margin, suggesting that the generated questions +are closely related to the interview process. Our model also per- +forms better than other baselines in Meaningfulness and Coherence +metrics, suggesting the overall higher quality of our model. +The above results demonstrate the competence of EZInterviewer +in producing meaningful and useful interview questions whilst +keeping the interview dialog flowing smoothly, just like a human +recruiter. Note that the average kappa statistics of human evaluation +are 0.51 and 0.48 on dialog level and interview level, respectively, +Figure 3: Visualization of key matching between dialog con- +text and selected resume keys, i.e., work experiment (Exp), +self description (Desc), skills (Ski), work years (Year), ex- +pected position (Pos), school (Sch), and major (Maj). 𝑈𝑖 de- +notes the 𝑖-th utterance. +which indicates moderate agreement between annotators. To prove +the significance of these results, we also conduct the two-tailed +paired student t-test between our model and DRD (row with shaded +background). The statistical significance of observed differences is +denoted using ▲(or ▼) for strong (or weak) significance for 𝛼 = 0.01. +Moreover, we obtain an average p-value of 5 × 10−6 and 3 × 10−4 +for both levels, respectively. +5.2 +Ablation Study +We conduct an ablation study to assess the contribution of individ- +ual components in the model. The results are shown in Table 2. +To verify the effectiveness of knowledge memory, we omit the +knowledge selection of dialog context history and directly use the +last utterance representation to select knowledge. The results (see +row w/o KM) confirm that employing each turn of historical dialog +to select knowledge and saving it in memory contribute to gener- +ating better responses. To confirm whether selecting knowledge +helps with the response generation process, we remove it from the +model, then simply add the representation of each utterance with +all resume values, and store it into the memory. This results in a +drop of 5.42% in BLEU-1 (see row w/o KS), suggesting that selecting +resume knowledge is beneficial in response generation. +5.3 +Analysis of Knowledge Selector +In Section § 3.4, we introduce the selecting mechanism of knowl- +edge selector, where the final attention (matching) score is obtained +in Equation 9. To study what specific information is attended by +the knowledge selector, and whether the selected information is +suitable for the next interview question, we conduct a case study to +visualize the matching score produced by the knowledge selector, +as shown in Table 4 and Figure 3. The first utterance in the history +is “Have you been engaged in front-end development work before?”, +and the knowledge selector learns that this utterance focuses on the +work experience in the resume. Accordingly, the fourth utterance “I +have more than 10 years of work experience.” pays more attention +to work years and work experience than other items in the resume. +This demonstrates that the knowledge selector learns which item +in the resume to focus on when generating each utterance. Hence, +when we want to ask the candidate to “introduce a React related +project”, the knowledge selector focuses on the work experience in +the resume and generates the mock interview question. + +U1 +U2 +U3 +U4 +Ski +Sch +Exp +Year +Desc +Pos +MajWSDM ’23, February 27-March 3, 2023, Singapore, Singapore +Mingzhe Li et al. +Table 4: Translated interview questions generated by baselines and EZInterviewer: an example. +denotes information +extracted and words generated by knowledge selector, whereas +denotes words generated from dialog generator. +Resume +Interview +Gender +Male +Job Description: +The main content of this work includes design and development based on the React +front-end framework. It requires the ability to efficiently complete front-end +development work and serve customers well. +Age +28 +Education +Undergraduate +Major +Computer Science +Work Years +10 +Context: +U1: Have you been engaged in front-end development work before? +U2: Yes, I am good at Vue, Node.js and some other skills. +U3: Okay, so do you have any React related experience? +U4: Yes, I have more than 10 years of work experience. +Expected Position +Front-end Engineer +Low Salary +5 +High Salary +6 +Skills +Vue, Node.js, Java +Experience +I was engaged in front-end design and was re- +sponsible for the project development based +on the React front-end framework and par- +ticipated in the system architecture process. +Ground Truth: So can you introduce a React related project you have done? +DDMN: What other front-end frameworks would you use? +DRD: Hello, can you tell us about your previous work? +EZInterviewer: Well, can you introduce the experience based on React framework? +Figure 4: Automatic evaluation metrics of DDMN, DRD and EZInterviewer on training data of different scales. +5.4 +Impact of Training Data Scales +To understand how our model and baseline models perform in a low- +resource scenario, we first evaluate them on the full training dataset, +then on smaller portions of the training dataset. Figure 4 presents +the performance of the models, DDMN, DRD, and EZInterviewer, +on the full, 1/2, 1/4, 1/8 and 1/10 of the training dataset (data scale), +respectively. It is observed that as the size of training dataset re- +duces, DDMN suffers a massive drop across all metrics, whereas the +scores of pre-training based models, i.e., DRD and EZInterviewer, +stay relatively stable. This demonstrates pre-training as an effective +strategy to tackle the low-resource challenge. Moreover, our model +outperforms DRD on all data scales, demonstrating the superiority +of our model. Figure 4 shows EZInterviewer eventually achieves the +best performance on all metrics and outperforms (albeit slightly), +with only 1/10 training data against all state-of-the-art baselines +trained with the full training dataset. +5.5 +Case Study +Table 4 presents a translated example of EZInterviewer and baseline +models. We observe that the question from EZInterviewer not only +catches the context, but also expands the conversation with proper +knowledge. This is highlighted in color codes: pink-colored words, +i.e., “experience” and “React framework”, are what knowledge selec- +tor extracts from resume knowledge, whereas blue-colored words, +i.e., “Well, can you introduce the...” and “based on”, which closely +connect to the context, are generated by dialog generator. In con- +trast, the questions from the baselines respond to the dialog but fail +to make connection with the resume knowledge. +6 +CONCLUSION +In this paper, we conduct a pilot study for the novel application of +intelligent online recruitment, namely EZInterviewer, which aims +to serve as mock interviewers for job-seekers. The mock interview +is generated with thorough understanding of the candidate’s re- +sume, the job requirements, the previous utterances in the context, +as well as the selected knowledge for grounded interviews. To ad- +dress the low-resource challenge, EZInterviewer is trained on a very +small set of interview dialogs. The key idea is to reduce the number +of parameters that rely on interview dialogs by disentangling the +knowledge selector and dialog generator so that most parameters +can be trained with ungrounded dialogs as well as the resume data +that are not low-resource. We conduct extensive experiments to +demonstrate the effectiveness of the proposed solution EZInter- +viewer. Our model achieves the best results using full training data +as well as small subsets of the training data in terms of various +metrics such as BLEU, embedding based similarity and diversity, +as well as human judgments. In particular, the human evaluation +indicates that our solution EZInterviewer can provide satisfactory +mock interviews to help the job-seekers prepare the real interview, +making the interview preparation process easier. +ACKNOWLEDGMENTS +We would like to thank the anonymous reviewers for their con- +structive comments. This work was supported by National Natural +Science Foundation of China (NSFC Grant No. 62122089). Rui Yan +is supported by Beijing Academy of Artificial Intelligence (BAAI). + +0.600 +0.575 +Score +0.550 +0.525 +BLEU +0.500 +DDMN +0.475 +DRD +0.450 +EZInterviewer +1 +1/2 +1/4 +1/8 +1/10 +Data Scale0.105 +: Score +0.100 +Distinct +0.095 +0.090 +DDMN +DRD +0.085 +EZinterviewer +i +1/2 +1/4 +1/8 +1/10 +Data ScaleEmbedding Score +0.78 +0.76 +DDMN +0.74 +DRD +EZInterviewer +0.72 +0.70 +i +1/2 +1/4 +1/8 +1/10 +Data Scale0.39 +0.38 +score +0.37 +S +0.36 +Entity +0.35 +0.34 +DDMN +DRD +0.33 +EZInterviewer +0.32 +i +1/2 +1/4 +1/8 +1/10 +Data ScaleScore +0.31 +0.30 +Correlation s +0.29 +0.28 +DDMN +DRD +0.27 +EZlnterviewer +i +1/2 +1/4 +1/8 +1/10 +Data ScaleEZInterviewer: To Improve Job Interview Performance with Mock Interview Generator +WSDM ’23, February 27-March 3, 2023, Singapore, Singapore +REFERENCES +[1] Martín Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey +Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, Manju- +nath Kudlur, Josh Levenberg, Rajat Monga, Sherry Moore, Derek Gordon Murray, +Benoit Steiner, Paul A. 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Difference- +aware Knowledge Selection for Knowledge-grounded Conversation Generation. +In Findings of the Association for Computational Linguistics: EMNLP 2020. 115–125. + diff --git a/0dAzT4oBgHgl3EQfDPpe/content/tmp_files/load_file.txt b/0dAzT4oBgHgl3EQfDPpe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..77169e026320dad5304fff4f6977c519e075c0ee --- /dev/null +++ b/0dAzT4oBgHgl3EQfDPpe/content/tmp_files/load_file.txt @@ -0,0 +1,882 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf,len=881 +page_content='EZInterviewer: To Improve Job Interview Performance with Mock Interview Generator Mingzhe Li∗† Peking University li_mingzhe@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='cn Xiuying Chen* CBRC, KAUST CEMSE, KAUST xiuying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='chen@kaust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='sa Weiheng Liao Made by DATA Liaoweiheng@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='com Yang Song BOSS Zhipin NLP Center songyang@kanzhun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='com Tao Zhang BOSS Zhipin kylen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='zhang@kanzhun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='com Dongyan Zhao Peking University zhaody@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='cn Rui Yan‡ Gaoling School of AI Renmin University of China ruiyan@ruc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='cn ABSTRACT Interview has been regarded as one of the most crucial step for recruitment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To fully prepare for the interview with the recruiters, job seekers usually practice with mock interviews between each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' However, such a mock interview with peers is generally far away from the real interview experience: the mock interviewers are not guaranteed to be professional and are not likely to behave like a real interviewer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Due to the rapid growth of online recruitment in recent years, recruiters tend to have online interviews, which makes it possible to collect real interview data from real interviewers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' In this paper, we propose a novel application named EZInterviewer, which aims to learn from the online interview data and provides mock interview services to the job seekers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The task is challenging in two ways: (1) the interview data are now available but still of low-resource;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' (2) to generate meaningful and relevant interview dialogs requires thorough understanding of both resumes and job descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To address the low-resource challenge, EZInterviewer is trained on a very small set of interview dialogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The key idea is to reduce the number of parameters that rely on interview dialogs by disentangling the knowledge selector and dialog generator so that most parameters can be trained with ungrounded dialogs as well as the resume data that are not low-resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Specifically, to keep the dialog on track for professional interviews, we pre-train a knowledge selector module to extract information from resume in the job-resume matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' A dialog generator is also pre-trained with ungrounded dialogs, learning to generate fluent responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Both authors contributed equally to this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' † Work done during an internship at BOSS Zhipin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' ‡ Corresponding author: Rui Yan (ruiyan@ruc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Permission to make digital or hard copies of all or part of this 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='1145/3539597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='3570476 Then, a decoding manager is finetuned to combine information from the two pre-trained modules to generate the interview ques- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Evaluation results on a real-world job interview dialog dataset indicate that we achieve promising results to generate mock in- terviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' With the help of EZInterviewer, we hope to make mock interview practice become easier for job seekers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' CCS CONCEPTS Computing methodologies → Natural language generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' KEYWORDS EZInterviewer, mock interview generation, knowledge-grounded dialogs, online recruitment, low-resource deep learning ACM Reference Format: Mingzhe Li, Xiuying Chen, Weiheng Liao, Yang Song, Tao Zhang, Dongyan Zhao, Rui Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' EZInterviewer: To Improve Job Interview Performance with Mock Interview Generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' In Proceedings of the Sixteenth ACM Inter- national Conference on Web Search and Data Mining (WSDM ’23), February 27-March 3, 2023, Singapore, Singapore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' ACM, New York, NY, USA, 9 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='1145/3539597.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='3570476 1 INTRODUCTION To make better preparations, job seekers practice mock interviews, which aims to anticipate interview questions and prepare them for what they might get asked in their real turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' However, the outcome of such an approach is unsatisfactory, since those “mock interviewers” do not have interview experience themselves, and do not know what the real recruiters would be interested in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Mock Interview Generation (MIG) represents a plausible solution to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Not only makes interviews more cost-effective, but mock interview generators also appear to be feasible, since much can be learned about the job seekers from their resumes, as can the job itself from the job description (JD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' An illustration of MIG task is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' There are two main challenges in this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' One is that the knowledge-grounded interviews are extremely time-consuming and costly to collect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Without a sufficient amount of training data, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='00972v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='CL] 3 Jan 2023 WSDM ’23, February 27-March 3, 2023, Singapore, Singapore Mingzhe Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Figure 1: An example of the Mock Interview Generation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Based on the candidate’s work experience and the current di- alog on the experience of web page development, the system generates an interview question “If a product needs a three- level classification selection, which component would you use and how to achieve it?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' the performance of such dialog generation models drops dramati- cally [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The second challenge is to make the knowledge-grounded dialog relevant to the candidate resume, job description, and previ- ous dialog utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' This makes MIG a complex task involving text understanding, knowledge selection, and dialog generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' In this paper, we propose EZInterviewer, a novel mock interview generator, with the aim of making interviews easier to prepare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The key idea is to train EZInterviewer in a low-resource setting: the model is first pre-trained on large-scale ungrounded dialogs and resume data, and then fine-tuned on a very small set of resume- grounded interview dialogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Specifically, the knowledge selector consists of a resume encoder to encode the resume, and a key-value memory network with mask self-attention mechanism, responsible for selecting relevant information in the resume to focus on to help generate the next interview utterance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The dialog generator also has two components, a context encoder which encodes the current dialog context, and a response decoder, responsible for generating the next dialog utterance without knowledge from the resumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' This knowledge-insensitive dialog generator is coordinated with the knowledge selector by a decoding manager that dynamically determines which component is activated for utterance generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' It is noted that the number of parameters in the decoding man- ager can be small, therefore it only requires a small number of resume-grounded interview dialogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Extensive experiments on real- world interview dataset demonstrate the effectiveness of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To summarize, our contributions are three-fold: We introduce a novel Mock Interview Generation task, which is a pilot study of intelligent online recruitment with potential commercial values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To address the low-resource challenge, we propose to reduce the number of parameters that rely on interview dialogs by dis- entangling knowledge selector and dialog generator so that the majority of parameters can be trained with large-scale ungrounded dialog and resume data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We propose a novel model to jointly process dialog contexts, candidate resumes, and job descriptions and generate highly rele- vant, knowledge-aware interview dialogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 2 RELATED WORK Multi-turn response generation aims to generate a response that is natural and relevant to the entire context, based on utterances in its previous turns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' [36] concatenated multiple utterances into one sen- tence and utilized RNN encoder or Transformer to encode the long sequence, simplifying multi-turn dialog into a single-turn dialog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To better model the relationship between multi-turn utterances, [4, 10] introduced interaction between utterances after encoding each utterance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' As human conversations are almost always grounded with exter- nal knowledge, the absence of knowledge grounding has become one of the major gaps between current open-domain dialog systems and real human conversations [8, 24, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' A series of work [20, 29] focused on generating a response based on the interaction between context and unstructured document knowledge, while a few oth- ers [22, 33] introduced knowledge graphs into conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' These models, however, usually under-perform in a low-resource setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To address the low resource problem, [16] proposed to enhance the context-dependent cross-lingual mapping upon the pre-trained monolingual BERT representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' [28] extended the meta-learning algorithm, which utilized knowledge learned from high-resource domains to boost the performance of low-resource unsupervised neural machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Different from the above methods, [37] proposed a disentangled response decoder in order to isolate pa- rameters that depend on knowledge-grounded dialogs from the entire generation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Our model takes a step further, taking into account the changes in attention on knowledge in multi-turn dialog scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 3 MODEL 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='1 Problem Formulation For an input multi-turn dialog context 𝑈 = {𝑢1,𝑢2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' ,𝑢𝑚} be- tween a job candidate and an interviewer, where 𝑢𝑖 represents the 𝑖-th utterance, we assume there is a ground truth textual interview question 𝑌 = {𝑦1,𝑦2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' ,𝑦𝑛}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 𝑚 is the utterance number in the dialog context and 𝑛 is the total number of words in question 𝑌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' In the 𝑖-th utterance, 𝑢𝑖 = {𝑥𝑖 1,𝑥𝑖 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' ,𝑥𝑖 𝑇 𝑖𝑢 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Meanwhile, there is a candidate resume 𝑅 = {(𝑘1, 𝑣1), (𝑘2, 𝑣2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' , (𝑘𝑇𝑟 , 𝑣𝑇𝑟 )} correspond- ing to the candidate in the interview, which has 𝑇𝑟 key-value pairs, and each of which represents an attribute in the resume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' For the job-resume matching pre-training task, there is an external job description 𝐽 = {𝑗1, 𝑗2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' , 𝑗𝑇𝑗 }, which has 𝑇𝑗 words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The goal is to generate an interview question 𝑌 ′ that is not only coherent with the dialog context 𝑈 but also pertinent to the job candidate’s resume 𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content="2 System Overview In this section, we propose our Low-resource Mock Interview Gen- erator (EZInterviewer) model, which is divided into three parts as shown in Figure 2: O 100%10:38 100%10:47 99%11:09 MyOnlineResume Preview < < Mock Interview Web Front-end Expected Position Hello, I'm an undergraduate, and I Development Engineer am confident that I am qualified for Python 8-9K Fresh graduate Undergraduate the intern position of web front-end San Francisco engineer." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='I hope you can see my information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' HR Work Experience Do you have experience in Job Description Company Emini programs or pc website 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='01-now> Web front-end development Web front-end development Webpack development?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' GIT Gulp JavaScript Vue :Iusedtodevelopfront-endwebpagesbasedor Adjust the style of the system :HTML and CSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=" I'm sorry that I have not done any back-stage on the PC front-end and call mini program work, but I used to Web front-end Vue Mini program develop Web page in the past." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=" the interface to decelop a small part of thefunction Work closely with the back-end devel- Project Experience :Let's do a test." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' If a product needs a opment team to ensure limited code :three-level .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' which component would you use :docking,optimize front-end perfor- and how to achieve it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Education Experience ④ :mance, and participate in mobile inter- :face development and architecture university 2019-2022 :design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Message ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Undergraduate Computer ScienceEZInterviewer: To Improve Job Interview Performance with Mock Interview Generator WSDM ’23, February 27-March 3, 2023, Singapore, Singapore Job Desc Job Encoder Resume Resume Encoder Multi-turn Interview History Self Attention Add & Norm Position-Wise FeedForward [CLS] [CLS] [CLS] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Utterance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='States ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Masked Self Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Cross Attention Manager ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Add & Norm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Position-Wise FeedForward ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Decoder Input ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='xN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Linear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Decoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='State ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Utterance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='States ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Cross ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Cross ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='xN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='updated output ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Fusion gate ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Pretrained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Soft Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Updated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Distribution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='One-hot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Target ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Masked ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Self-attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Visible Matrix ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Cross ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Transfer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Memory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Job-resume ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Matching ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Knowledge Selector (Pretraining) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Dialog Generator (Pretraining) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Decoding Manager ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Self Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Key-Value ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Memory Network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='Figure 2: Overview of EZInterviewer,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' which consists of three parts: (1) Knowledge Selector selects salient knowledge infor- mation from the candidate resume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' (2) Dialog Generator predicts the next word without knowledge of resumes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' (3) Decoding Manager coordinates the output from knowledge selector and dialog generator to produce the interview question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Dialog Generator predicts the next word of a response based on the prior sub-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' In our model, we pre-train it by large-scale ungrounded dialogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Knowledge Selector selects salient knowledge information from the candidate resume for interview question generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' In our model, we augment the ability of the knowledge selector by em- ploying it to perform job-resume matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Decoding Manager coordinates the output from knowledge selector and dialog generator to predict the interview question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' It is important to note that to train an EZInterviewer model, two pre-train techniques are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Firstly, we pre-train the knowledge selector in a job-matching task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' This is because while it is hard to attend to appropriate content in a resume just on its own, the salient information in a resume can be identified in a job-resume matching task [13, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Secondly, the context encoder and response decoder of the dialog generator are pre-trained with a large scale of ungrounded dialogs, so as to predict the next word of response based on the prior sub-sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Finally, the decoding manager, which relies on a few parameters, coordinates the two components to generate knowledge grounded interview utterance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='3 Dialog Generator Context Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Instead of processing the dialog context as a flat sequence, we employ a hierarchical encoder [3] to capture intra- and inter-utterance relations, which is composed of a local sentence encoder and a global context encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' For the sentence encoder, to model the semantic meaning of the dialog context, we learn the representation of each utterance 𝑢𝑖 by a self-attention mechanism (SAM) initialized by BERT [5]: ℎ𝑖 𝑗 = SAMu(𝑒(𝑥𝑖 𝑗),ℎ𝑖 ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' (1) We extract the state at “[cls]” position to denote the utterance state, abbreviated as ℎ𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Apart from the local information exchange in each utterance, we let information flow across multi-turn context: ℎ𝑐 𝑡 = SAMc(ℎ𝑡,ℎ𝑐 ∗), (2) where ℎ𝑐 𝑡 denotes the hidden state of the 𝑡-th utterance in SAMc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Response Decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Response decoder is responsible for under- standing the previous dialog context and generates the response without the knowledge of resume information [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Our decoder also follows the style of Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Concretely, we first apply the self-attention on the masked de- coder input, obtaining𝑑𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Based on𝑑𝑡 we compute the cross-attention scores over previous utterances: 𝛼𝑐 𝑡 = ReLU([𝑑𝑡𝑊𝑑 (ℎ𝑐 𝑖𝑊ℎ)𝑇 ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' (3) The attention weights 𝛼𝑐 𝑡 is then used to obtain the context vectors as𝑐𝑡 = �𝑚 𝑖=1 𝛼𝑐 𝑡 ℎ𝑐 𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The context vectors𝑐𝑡, treated as salient contents of various sources, are concatenated with the decoder hidden state 𝑑𝑡 to produce the distribution over the target vocabulary: 𝑃𝑤 𝑣 = Softmax (𝑊𝑜 [𝑑𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='𝑐𝑡]) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' (4) Pre-training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' While interview dialogs are hard to come by, online conversation is abundant on the internet, and can be easily collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Hence, we pre-train the dialog generator on un- grounded conversations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Concretely, during pre-training process, we employ the context encoder to first encode the multi-turn pre- vious dialog context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Then, at the 𝑡-th decoding step, we use the response decoder to predict the 𝑡-th word in the response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We set the loss as the negative log likelihood of the target word 𝑦𝑡: 𝐿𝑜𝑠𝑠𝑔 = − 1 𝑛 �𝑛 𝑡=1 log 𝑃𝑤 𝑣 (𝑦𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' (5) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='4 Knowledge Selector Resume Encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' As shown in Figure 2, a resume contains several key-value pairs (𝑘𝑖, 𝑣𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Most of key and value fields include a single word or a phrase such as “skills” or “gender”, and we can obtain the feature representation through an embedding matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Concretely, for each key or value field with a single word or a phrase, we estab- lish a corresponding resume embedding matrix 𝑒𝑖𝑟 that is different from the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Then we use the resume embedding matrix to map each field word 𝑘𝑖 or 𝑣𝑖 into to a high-dimensional vector space, denoted as 𝑒𝑖𝑟 (𝑘𝑖) or 𝑒𝑖𝑟 (𝑣𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' For fields with more than one word such as “work experience” or “I used to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='..”, we denote them as 𝑣𝑖 = (𝑣1 𝑖 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='𝑣𝑙𝑖 𝑖 ), where 𝑙𝑖 denotes the word number of the current Birthday 19980501 Gender MaleWSDM ’23, February 27-March 3, 2023, Singapore, Singapore Mingzhe Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We first process them through the previous word embedding matrix 𝑒, then there is an SAMR, similar with SAMu in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='3, to model the temporal interactions between words: ℎ𝑟𝑖 𝑡 = SAMR(𝑒(𝑣 𝑗 𝑖 ),ℎ𝑟𝑖 𝑡−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' (6) We use the last hidden state of the SAMR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=', ℎ𝑟𝑖 𝑙𝑖 to denote the overall representation for field 𝑣𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' For brevity, in the following sections, we use ℎ𝑘 𝑖 and ℎ𝑣 𝑖 to denote the encoded key-value pair (𝑘𝑖, 𝑣𝑖) in the resume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Masked Self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Traditional self-attention can be used to update representation of each resume item due to its flexibility in relating two elements in a distance-agnostic manner [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' However, as shown in [21], too much knowledge incorporation may divert the representation from its correct meaning, which is called knowledge noise (KN) issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' In our scenario, the information in the resume is divided into several parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=', basic personal information, work experiences and extended work, each of which contains variable number of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The items within each part are closely connected, while different parts can be considered as different domains, and the interaction may introduce a certain amount of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To over- come this problem, we introduce a visible matrix, in which items belonging to the same part are visible to each other, while the visi- bility degree between items is determined by the cosine similarity of semantic representations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=', 𝐶𝑖,𝑗 = cos_sim(ℎ𝑣 𝑖 ,ℎ𝑣 𝑗 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Then, the scaled dot-product masked self-attention is defined as: 𝛼𝑖,𝑗 = exp � (ℎ𝑘 𝑖 𝑊𝑞)𝐶𝑖,𝑗 (ℎ𝑘 𝑗𝑊𝑘)𝑇 � �𝑇𝑟 𝑛=1 exp � (ℎ𝑘 𝑖 𝑊𝑞)𝐶𝑖,𝑛(ℎ𝑘 𝑗𝑊𝑘)𝑇 � , (7) ˆℎ𝑣 𝑖 = ∑︁𝑇𝑟 𝑗=1 𝛼𝑖,𝑗ℎ𝑣 𝑗 √ 𝑑 , (8) where 𝑑 stands for hidden dimension and 𝐶 is the visible matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' ˆℎ𝑣 𝑖 is then utilized as the updated resume value representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Key-Value Memory Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The goal of key matching is to calculate the relevance between each attribute of the resume and the previous dialog context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Given dialog context ℎ𝑖, for the 𝑗-th attribute pair (𝑘𝑗, 𝑣𝑗), we calculate the probability of ℎ𝑖 over 𝑘𝑗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=', 𝑃(𝑘𝑗 |ℎ𝑖), as the matching score 𝛽𝑖,𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To this end, we exploit the context representation ℎ𝑖 to calculate the matching score: 𝛽𝑖,𝑗 = exp � ℎ𝑖𝑊𝑎ℎ𝑘 𝑗 � �𝑇𝑟 𝑛=1 exp � ℎ𝑖𝑊𝑎ℎ𝑘𝑛 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' (9) Since context representation ℎ𝑖 and resume key representation ℎ𝑘 𝑗 are not in the same semantic space, we use a trainable key matching parameter𝑊𝑎 to transform these representations into a same space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' As the relevance between context ℎ𝑖 and each pair in the resume table (𝑘𝑗, 𝑣𝑗), the matching score 𝛽𝑖,𝑗 can help to capture the most relevant pair for generating a correct question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Therefore, as shown in Equation 10, the knowledge selector reads the information 𝑀𝑖 from KVMN via summing over the stored values, and guides the follow-up response generation, so we have: 𝑀𝑖 = ∑︁𝑇𝑟 𝑗=1 𝛽𝑖,𝑗 ˆℎ𝑣 𝑗, (10) where ˆℎ𝑣 𝑗 is the representation of value 𝑣𝑗, and 𝛽𝑖,𝑗 is the matching score between dialog context ℎ𝑖 and key 𝑘𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Pre-training Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' In practice, the resume knowledge con- tains a variety of professional and advanced scientific concepts such as “Web front-end”, “HTML”, and “CSS”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' These technical terms are difficult to understand for people not familiar with the specific domain, not to mention for the model that is not able to access a large-scale resume-grounded dialog dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Hence, it would be difficult for the knowledge selector to understand the resume con- tent and previous context about the resume, so as to select the next resume pair to focus on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' On the other hand, we notice that in job-resume matching task, it is crucial to capture the decisive information in the resume to perform a good matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' For example, recruiters may tend to hire the candidate with particular experiences among several candidates with similar backgrounds [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Intuitively, the key-value pair that is important for job-resume matching is also the key factor to consider in a job interview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Hence, if we can let the model learn the salient information in the resume by performing the job-resume matching task on large-scale job-resume data, then it would also bring benefits for selecting salient information in interview question generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Concretely, we use the job description to attend to the resume to perform a job-resume matching task, as a pre-training process for knowledge selector module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' As shown in Figure 2, the Job Encoder encodes the job description by a SAMjd: ℎ𝑗𝑑 𝑖 = SAMjd(𝑒(𝑗𝑖),ℎ𝑗𝑑 𝑖−1), (11) where 𝑗𝑖 denotes the 𝑖-th word in the job description, and 𝑒(𝑗𝑖) is mapped by the previous embedding matrix 𝑒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We use the final hidden state of the SAMjd, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=', ℎ𝑗𝑑 𝑇𝑗 as the overall representation for the description, abbreviated as ℎ𝑗𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' ℎ𝑗𝑑 plays a similar part as the context representation ℎ𝑖, which first attends to the keys in the resume, and then is used to “weightedly” read the values in the resume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We use 𝑚𝑗𝑑 to denote the weighted read result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' In the training process, we first pre-train the knowledge selector by job-resume matching task, which can be formulated as a classi- fication problem [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The objective is to maximize the scores of positive samples while minimizing that of the negative samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Specifically, we concatenateℎ𝑗𝑑 and𝑚𝑗𝑑 since vector concatenation for matching is known to be effective [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Then the concatenated vector is fed to a multi-layer, fully-connected, feed-forward neural network, and the job-resume matching score 𝑠𝑗𝑟 is obtained as: 𝑠𝑗𝑟 = 𝜎 � 𝐹𝑠 ([ℎ𝑗𝑑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='𝑚𝑗𝑑]]) � , (12) where [;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' ] denotes concatenation operation, and the outputs are the probabilities of successfully matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We use the job-resume pairs in interviews as positive samples, and then use the job-resume pairs without interviews as negative instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' After pre-training, the job description is replaced by the context representations, while the key matching and value combination processes remain the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We use a knowledge memory 𝑀 to store the selection result, where each slot stores the value combination result 𝑀𝑖 in Equation 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' EZInterviewer: To Improve Job Interview Performance with Mock Interview Generator WSDM ’23, February 27-March 3, 2023, Singapore, Singapore 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='5 Decoding Manager The decoding manager is supposed to generate the proper word based on the knowledge memory and the response decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Our idea is inspired by an observation on the nature of interview dialogs: despite the fact that a dialog is based on the resume, words and utter- ances in the dialog are not always related to resume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Therefore, we postulate that formation of a response can be decomposed into two uncorrelated actions: (1) selecting a word according to the context to make the dialog coherent (corresponding to the dialog generator);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' (2) selecting a word according to the extra knowledge memory to ground the dialog (corresponding to the knowledge selector).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The two actions can be independently performed, which becomes the key reason why the large resume-job matching and ungrounded dialog datasets, although seemingly unrelated to interview dialogs, can be very useful in an MIG task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Note that in Section §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='4, we store the selected knowledge 𝑀𝑖 in a knowledge memory 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To select a word based on it, similar to the response decoder, we use 𝑑𝑡 to attend to each slot of knowledge memory, and we can obtain the knowledge context vector 𝑔𝑘 𝑡 and the output decoder state 𝑑𝑘𝑜 𝑡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The response decoder and knowledge selector are controlled by the decoding manager with a “fusion gate” to decide how much information from each side should be focused on at each step of interview question prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 𝛾𝑓 = 𝜎 (𝐹𝑚(𝑑𝑡)) , (13) where 𝑑𝑡 is the 𝑡-th decoder hidden state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Then, the probability to predict word 𝑦𝑡 can be formulated as: 𝑑𝑜 𝑡 = 𝛾𝑓 𝑑𝑤𝑜 𝑡 + (1 − 𝛾𝑓 )𝑑𝑘𝑜 𝑡 , (14) 𝑃𝑣 = softmax �𝑊𝑣𝑑𝑜 𝑡 + 𝑏𝑣 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' (15) As for the optimization goal, generation models that use one- hot distribution optimization target always suffer from the over- confidence issue, which leads to poor generation diversity [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Hence, aside from the ground truth one-hot label 𝑃, we also propose a soft target label 𝑃𝑤 𝑣 (see Equation 4), which is borrowed from the pre-trained Dialog Generator in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Forcing the decoding manager to simulate the pre-trained decoder can help it learn the context of the interview dialog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We combine the one-hot label with the soft label by an editing gate 𝜆, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Concretely, a smooth target distribution 𝑃 ′ is proposed to replace the hard target distribution 𝑃 as: 𝑃 ′ = 𝜆𝑃 + (1 − 𝜆)𝑃𝑤 𝑣 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' (16) where 𝜆 ∈ [0, 1] is an adaption factor, 𝑃𝑤 𝑣 is obtained from Equa- tion 4, and 𝑃 is the hard target as one-hot distribution which assigns a probability of 1 for the target word 𝑦𝑡 and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 4 EXPERIMENTAL SETUP 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='1 Dataset In this paper, we conduct experiments on a real-world dataset pro- vided by “Boss Zhipin” 1, the largest online recruiting platform in China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To protect the privacy of candidates, user records are anonymized with all personal identity information removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='zhipin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='com Table 1: Statistics of the datasets used in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Statistics Values Interview Dialog Dataset Total number of resumes 12,666 Total number of dialog utterances 49,214 Avg turns # per dialog context 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='47 Avg words # per utterance 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='18 Job-Resume Dataset Key-value pairs # per resume 22 Avg words # per work experience in resume 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='80 Avg words # per self description in resume 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='13 Avg words # per job description 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='26 Ungrounded Dialog Dataset Total number of context-response pairs 2,995,000 Avg turns # per dialog context 4 Avg words # per utterance 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='15 dataset includes 12,666 resumes, 8,032 job descriptions, and 49,214 interview dialog utterances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The statistics of the dataset is summa- rized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We then tokenize each sentence into words with the benchmark Chinese tokenizer toolkit “JieBa” 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To pre-train the knowledge selector module, we use a job-resume matching dataset [34], again from “Boss Zhipin”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The training set and the validation set include 355,000 and 1,006 job-resume pairs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To pre-train dialog generator, we choose Weibo dataset [2], which includes a massive number of multi-turn con- versations collected from “Weibo”3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The data includes 2,990,000 context-response pairs for training and 5,000 pairs for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The details are also summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='2 Comparisons We compare our proposed model against traditional knowledge- insensitive dialog generation baselines, and knowledge-aware dia- log generation baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Knowledge-insensitive dialog generation baselines: Transformer [30]: is based solely on attention mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' BERT [5]: initializes Transformer with BERT as the encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Di- aloGPT [36]: proposes a large, tunable neural conversational re- sponse generation model trained on more conversation-like ex- changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' T5-CLAPS [14]: generates samples for contrastive learn- ing by adding small and large perturbations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Knowledge-aware dialog generation baselines: TMN [6]: is built upon a transformer architecture with an ex- ternal memory hosting the knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' ITDD [20]: incrementally encodes multi-turn dialogs and knowledge and decodes responses with a deliberation technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' DiffKS [38]: utilizes the differential information between selected knowledge in multi-turn conversa- tion for knowledge selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' DRD [37]: tackles the low-resource challenge with pre-training techniques using ungrounded dialogs and documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' DDMN [31]: dynamically keeps track of dialog context for multi-turn interactions and incorporates KB knowledge 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='com/fxsjy/jieba 3https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='weibo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='com WSDM ’23, February 27-March 3, 2023, Singapore, Singapore Mingzhe Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Table 2: Comparing model performance on full dataset: automatic 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Persona [9]: introduces personal memory into knowledge selection to address the personalization issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='3 Implementation Details We implement our experiments in TensorFlow [1] on an NVIDIA GTX 1080 Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' For our model and all baselines, we follow the same setting as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We truncate input dialog to 100 words with 20 words in each utterance, as we did not find significant improvement when increasing input length from 100 to 200 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The minimum decoding step is 10, and the maximum step is 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The word embedding dimension is set to 128 and the number of hidden units is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Experiments are performed with a batch size of 256, and the vocabulary is comprised of the most frequent 50k words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We use Adam optimizer [12] as our optimizing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We selected the 5 best checkpoints based on performance on the validation set and report averaged results on the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Note that for better performance, our model is built based on BERT, and the decoding process is the same as Transformer [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Finally, due to the limitation of time and memory, small settings are used in the pre-trained baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='4 Evaluation Metrics To evaluate the performance of EZInterviewer against baselines, we adopt the following metrics widely used in existing studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Overlap-based Metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Following [18], we utilize BLEU score [25] to measure n-grams overlaps between ground-truth and gener- ated response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' In addition, we apply Correlation (Cor) to calculate the words overlap between generated question and job description, which measures how well the generated questions line up with the recruitment intention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Embedding Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We compute the similarity between the bag-of-words (BOW) embeddings of generated results and reference to capture their semantic matching degrees [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' In particular we adopt three metrics: 1) Greedy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=', greedily matching words in two Table 3: Human evaluation results on: Readability (Read), Informativeness (Info), Meaningfulness (Mean), Usefulness (Use), Relevance (Rel), and Coherence (Coh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Model Dialog-level Interview-level Read Info Mean Use Rel Coh DiffKS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='79 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='87 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='03 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='99 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='10 DDMN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='97 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='63 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='91 DRD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='09 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='02 EZInterviewer 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='42▲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='51▲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='39▲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='46▲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='57▲ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='38▲ utterances based on cosine similarities;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 2) Average, cosine similarity between the averaged word embeddings in two utterances [23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 3) Extrema, cosine similarity between the largest extreme values among the word embeddings in the two utterances [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Distinctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The distinctness score [15] measures word-level diversity by calculating the ratio of distinct uni-gram and bi-grams in generated responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Entity F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Entity F1 is computed by micro-averaging precision and recall over knowledge-based entities in the entire set of sys- tem responses, and evaluates the ability of a model to generate relevant entities to achieve specific tasks from the provided knowl- edge base [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The entities we use are extracted from an entity vocabulary provided by “Boss Zhipin”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Human Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We further employ human eval- uations aside from automatic evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Three well-educated annotators from different majors are hired to evaluate the quality of generated responses, where the evaluation is conducted in a double-blind fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' In total 100 randomly sampled responses gen- erated by each model are rated by each annotator on both dialog level and interview level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We adopt the Readability (is the response grammatically correct?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=') and Informativeness (does the response include informative words?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=') to judge the quality of the generated EZInterviewer: To Improve Job Interview Performance with Mock Interview Generator WSDM ’23, February 27-March 3, 2023, Singapore, Singapore responses on the dialog level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' On the interview level, we adopt Meaningfulness (is the generated question meaningful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' ), Usefulness (is the question worth the job candidate preparing in advance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' ), Relevance (is the question relevant to the resume?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=') and Coherence (is the generated text coherent with the context?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=') to assess the overall performance of a model and the quality of user experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Each metric is given a score between 1 and 3 (1 = bad, 2 = average, 3 = good).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 5 EXPERIMENTAL RESULT 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='1 Overall Performance Automatic evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The comparison between EZInterviewer and state-of-the-art generative baselines is listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We take note that the knowledge-aware dialog generation mod- els outperform traditional dialog models, suggesting that utilizing external knowledge introduces advantages in generating relevant response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We also notice the pre-train based model DRD outper- forms other baselines, showing that initializing parameters by pre- training on large-scale data can lead to a substantial improvement in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' It is worth noting some models achieve better En- tity F1 but a lower BLEU score;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' this suggests that those models tend to copy necessary entity words from the knowledge but are not able to use them properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' EZInterviewer outperforms baselines on all automatic metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Firstly, our model improves BLEU-1 by 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='92% over DRD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' On the Dis- tinctness metric Dist-1, our model outperforms DialoGPT by 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='99%, suggesting that the generated interview questions are diversified and personalized with different candidates’ resumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Moreover our model attains a good score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='3927 on entity F1, which evaluates the degree to which the generated question is grounded on the knowledge base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Finally, Cor score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='3145 suggests the ques- tions generated by EZInterviewer is in line with the job description, hence reflect the intention of the recruiters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Overall the metrics demonstrate that our model successfully learns an interviewer’s points of interest in a resume, and incorporates this knowledge into interview questions properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Human evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The results of human evaluations on all models are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' EZInterviewer is the top performer on all the metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Specifically, our model outperforms DiffKS by 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='20% on Readability, suggesting that EZInterviewer manages to reduce the grammatical errors and improve the readability of the generated response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' As for the Informativeness metric, our model scores 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='68 higher than DDMN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' This indicates that EZInterviewer captures salient information in the resume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' On the interview level, EZInter- viewer’s Usefulness score is 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='27% better than DRD, demonstrating its capabilities to help job seekers to pick the right questions to prepare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' On Relevance metric, our model outperforms all baselines by a considerable margin, suggesting that the generated questions are closely related to the interview process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Our model also per- forms better than other baselines in Meaningfulness and Coherence metrics, suggesting the overall higher quality of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The above results demonstrate the competence of EZInterviewer in producing meaningful and useful interview questions whilst keeping the interview dialog flowing smoothly, just like a human recruiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Note that the average kappa statistics of human evaluation are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='51 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='48 on dialog level and interview level, respectively, Figure 3: Visualization of key matching between dialog con- text and selected resume keys, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=', work experiment (Exp), self description (Desc), skills (Ski), work years (Year), ex- pected position (Pos), school (Sch), and major (Maj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 𝑈𝑖 de- notes the 𝑖-th utterance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' which indicates moderate agreement between annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To prove the significance of these results, we also conduct the two-tailed paired student t-test between our model and DRD (row with shaded background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The statistical significance of observed differences is denoted using ▲(or ▼) for strong (or weak) significance for 𝛼 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Moreover, we obtain an average p-value of 5 × 10−6 and 3 × 10−4 for both levels, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='2 Ablation Study We conduct an ablation study to assess the contribution of individ- ual components in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The results are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To verify the effectiveness of knowledge memory, we omit the knowledge selection of dialog context history and directly use the last utterance representation to select knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The results (see row w/o KM) confirm that employing each turn of historical dialog to select knowledge and saving it in memory contribute to gener- ating better responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To confirm whether selecting knowledge helps with the response generation process, we remove it from the model, then simply add the representation of each utterance with all resume values, and store it into the memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' This results in a drop of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='42% in BLEU-1 (see row w/o KS), suggesting that selecting resume knowledge is beneficial in response generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='3 Analysis of Knowledge Selector In Section § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='4, we introduce the selecting mechanism of knowl- edge selector, where the final attention (matching) score is obtained in Equation 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To study what specific information is attended by the knowledge selector, and whether the selected information is suitable for the next interview question, we conduct a case study to visualize the matching score produced by the knowledge selector, as shown in Table 4 and Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The first utterance in the history is “Have you been engaged in front-end development work before?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=', and the knowledge selector learns that this utterance focuses on the work experience in the resume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Accordingly, the fourth utterance “I have more than 10 years of work experience.” pays more attention to work years and work experience than other items in the resume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' This demonstrates that the knowledge selector learns which item in the resume to focus on when generating each utterance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Hence, when we want to ask the candidate to “introduce a React related project”, the knowledge selector focuses on the work experience in the resume and generates the mock interview question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' U1 U2 U3 U4 Ski Sch Exp Year Desc Pos MajWSDM ’23, February 27-March 3, 2023, Singapore, Singapore Mingzhe Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Table 4: Translated interview questions generated by baselines and EZInterviewer: an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' denotes information extracted and words generated by knowledge selector, whereas denotes words generated from dialog generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Resume Interview Gender Male Job Description: The main content of this work includes design and development based on the React front-end framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' It requires the ability to efficiently complete front-end development work and serve customers well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Age 28 Education Undergraduate Major Computer Science Work Years 10 Context: U1: Have you been engaged in front-end development work before?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' U2: Yes, I am good at Vue, Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='js and some other skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' U3: Okay, so do you have any React related experience?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' U4: Yes, I have more than 10 years of work experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Expected Position Front-end Engineer Low Salary 5 High Salary 6 Skills Vue, Node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='js, Java Experience I was engaged in front-end design and was re- sponsible for the project development based on the React front-end framework and par- ticipated in the system architecture process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Ground Truth: So can you introduce a React related project you have done?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' DDMN: What other front-end frameworks would you use?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' DRD: Hello, can you tell us about your previous work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' EZInterviewer: Well, can you introduce the experience based on React framework?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Figure 4: Automatic evaluation metrics of DDMN, DRD and EZInterviewer on training data of different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='4 Impact of Training Data Scales To understand how our model and baseline models perform in a low- resource scenario, we first evaluate them on the full training dataset, then on smaller portions of the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Figure 4 presents the performance of the models, DDMN, DRD, and EZInterviewer, on the full, 1/2, 1/4, 1/8 and 1/10 of the training dataset (data scale), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' It is observed that as the size of training dataset re- duces, DDMN suffers a massive drop across all metrics, whereas the scores of pre-training based models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=', DRD and EZInterviewer, stay relatively stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' This demonstrates pre-training as an effective strategy to tackle the low-resource challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Moreover, our model outperforms DRD on all data scales, demonstrating the superiority of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Figure 4 shows EZInterviewer eventually achieves the best performance on all metrics and outperforms (albeit slightly), with only 1/10 training data against all state-of-the-art baselines trained with the full training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='5 Case Study Table 4 presents a translated example of EZInterviewer and baseline models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We observe that the question from EZInterviewer not only catches the context, but also expands the conversation with proper knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' This is highlighted in color codes: pink-colored words, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=', “experience” and “React framework”, are what knowledge selec- tor extracts from resume knowledge, whereas blue-colored words, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=', “Well, can you introduce the.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='..” and “based on”, which closely connect to the context, are generated by dialog generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' In con- trast, the questions from the baselines respond to the dialog but fail to make connection with the resume knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 6 CONCLUSION In this paper, we conduct a pilot study for the novel application of intelligent online recruitment, namely EZInterviewer, which aims to serve as mock interviewers for job-seekers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The mock interview is generated with thorough understanding of the candidate’s re- sume, the job requirements, the previous utterances in the context, as well as the selected knowledge for grounded interviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' To ad- dress the low-resource challenge, EZInterviewer is trained on a very small set of interview dialogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' The key idea is to reduce the number of parameters that rely on interview dialogs by disentangling the knowledge selector and dialog generator so that most parameters can be trained with ungrounded dialogs as well as the resume data that are not low-resource.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' We conduct extensive experiments to demonstrate the effectiveness of the proposed solution EZInter- viewer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Our model achieves the best results using full training data as well as small subsets of the training data in terms of various metrics such as BLEU, embedding based similarity and diversity, as well as human judgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' In particular, the human evaluation indicates that our solution EZInterviewer can provide satisfactory mock interviews to help the job-seekers prepare the real interview, making the interview preparation process easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' ACKNOWLEDGMENTS We would like to thank the anonymous reviewers for their con- structive comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' This work was supported by National Natural Science Foundation of China (NSFC Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 62122089).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Rui Yan is supported by Beijing Academy of Artificial Intelligence (BAAI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='575 Score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='525 BLEU 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='500 DDMN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='475 DRD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='450 EZInterviewer 1 1/2 1/4 1/8 1/10 Data Scale0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='105 : Score 0.' metadata={'source': 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+page_content='36 Entity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='34 DDMN DRD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='33 EZInterviewer 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='32 i 1/2 1/4 1/8 1/10 Data ScaleScore 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='30 Correlation s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='28 DDMN DRD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='27 EZlnterviewer i 1/2 1/4 1/8 1/10 Data ScaleEZInterviewer: To Improve Job Interview Performance with Mock Interview Generator WSDM ’23, February 27-March 3, 2023, Singapore, Singapore REFERENCES [1] Martín Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, Manju- nath Kudlur, Josh Levenberg, Rajat Monga, Sherry Moore, Derek Gordon Murray, Benoit Steiner, Paul A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Tucker, 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' Dialogpt: Large-scale generative pre-training for conversational response generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' arXiv preprint arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content='00536 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' [37] Xueliang Zhao, Wei Wu, Chongyang Tao, Can Xu, Dongyan Zhao, and Rui Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} +page_content=' 115–125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfDPpe/content/2301.00972v1.pdf'} diff --git a/0dE4T4oBgHgl3EQfZgyj/content/tmp_files/2301.05057v1.pdf.txt b/0dE4T4oBgHgl3EQfZgyj/content/tmp_files/2301.05057v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..715104e0e24aceb14202697d933e68a0f1ec585b --- /dev/null +++ b/0dE4T4oBgHgl3EQfZgyj/content/tmp_files/2301.05057v1.pdf.txt @@ -0,0 +1,1463 @@ +arXiv:2301.05057v1 [q-bio.QM] 19 Dec 2022 +AN OVERVIEW OF OPEN SOURCE DEEP LEARNING-BASED +LIBRARIES FOR NEUROSCIENCE +Louis Fabrice Tshimanga +Department of Neuroscience (DNS) +University of Padova +louisfabrice.tshimanga@unipd.it +Manfredo Atzori +Department of Neuroscience (DNS), +Padova Neuroscience Center (PNC) +University of Padova +Information Systems Institute +University of Applied Sciences Western Switzerland (HES-SO Valais) +manfredo.atzori@unipd.it +Federico Del Pup +Department of Neuroscience (DNS), +Department of Information Engineering (DEI) +University of Padova +federico.delpup@studenti.unipd.it +Maurizio Corbetta +Department of Neuroscience (DNS), +Padova Neuroscience Center (PNC) +University of Padova +Department of Neurology +Washington University School of Medicine +maurizio.corbetta@unipd.it +ABSTRACT +In recent years, deep learning revolutionized machine learning and its applications, producing re- +sults comparable to human experts in several domains, including neuroscience. Each year, hundreds +of scientific publications present applications of deep neural networks for biomedical data analysis. +Due to the fast growth of the domain, it could be a complicated and extremely time-consuming task +for worldwide researchers to have a clear perspective of the most recent and advanced software +libraries. This work contributes to clarify the current situation in the domain, outlining the most +useful libraries that implement and facilitate deep learning application to neuroscience, allowing +scientists to identify the most suitable options for their research or clinical projects. This paper +summarizes the main developments in Deep Learning and their relevance to Neuroscience; it then +reviews neuroinformatic toolboxes and libraries, collected from the literature and from specific hubs +of software projects oriented to neuroscience research. The selected tools are presented in tables +detailing key features grouped by domain of application (e.g. data type, neuroscience area, task), +model engineering (e.g. programming language, model customization) and technological aspect +(e.g. interface, code source). The results show that, among a high number of available software +tools, several libraries are standing out in terms of functionalities for neuroscience applications. The +aggregation and discussion of this information can help the neuroscience community to devolop +their research projects more efficiently and quickly, both by means of readily available tools, and by +knowing which modules may be improved, connected or added. +Keywords Deep Learning · Neuroscience · Neuroinformatics · Open source +1 +Introduction +In the last decade, Deep Learning (DL) has taken over most classic approaches in Machine Learning (ML), Computer +Vision, Natural Language Processing, showing an unprecedented versatility, and matching or surpassing the perfor- +mances of human experts in narrow tasks. +The recent growth of DL applications to several domains, including Neuroscience, consequently offers numerous open- + +source software opportunities for researchers. +Mapping available resources can allow a faster and more precise exploitation. +Neuroscience is a diversified field on its own, as much for the objects and scales it focuses on, as for the types of data +it relies on. +The discipline is also historically tied to developments in electrical, electronic, and information technology. Modern +Neuroscience relies on computerization in many aspects of data generation, acquisition, and analysis. Statistical and +Machine Learning techniques already empower many software packages, that have become de facto standards in sev- +eral subfields of Neuroscience, such as Principal and Independent Component Analysis in Electroencephalography +and Neuroimaging, to name a few. +Meanwhile, the rich and rapidly evolving taxonomy of Deep Neural Networks (DNNs) is becoming both an opportu- +nity and hindrance. On the one hand, currently open-source DL libraries allow an increasing number of applications +and studies in Neuroscience. On the other hand, the adoption of available methods is slowed down by a lack of stan- +dards, reference frameworks and established workflows. Scientific communities whose primary focus or background +is not in machine learning engineering may be left partially aside from the ongoing Artificial Intelligence (AI) gold +rush. +For such reasons it is fundamental to overview open-source libraries and toolkits. Framing a panorama could help +researchers in selecting ready-made tools and solutions when convenient, as well as in pointing out and filling in the +blanks with new applications. This work would contribute to advancing the community’s possibilities, reducing the +workload for researchers to exploit DL, allowing Neuroscience to benefit of its most recent advancements. +2 +Background +2.1 +Deep Learning +Deep Learning (DL) has contributed many best solutions to problems in its parent field, Machine Learning, thanks to +theoretical and technological achievements that unlocked its intrinsic versatility. +Machine Learning is the study of computer algorithms that tackle problems without complete access to predefined +rules or analytical, closed-form solutions. +The algorithms often require a training phase to adjust parameters and satisfy internal or external constraints (e.g. of +exactness, approximation or generality) on dedicated data for which solutions might be already known. +Machine Learning comprises a wide array of statistical and mathematical methods, including Artificial Neural Net- +works (ANNs), biologically inspired systems that connect inputs and outputs through simple computing units (neu- +rons), which act as function approximators. +Each unit implements a nonlinear function of the weighted sum of its inputs, thus the output of the whole ANN is a +composite function, as formally intended in mathematics. The networks of neurons are most often layered and "feed- +forward", meaning that units from any layer only output results to units in subsequent layers. The width of a layer +refers to its neuron count, while the depth of a network refers to its layer count. The typical architecture instantiating +the above characteristics is the MultiLayer Perceptron [1] (MLP). +Universal approximation theorems [2] [3] ensure that, whenever a nonlinear network as the MLP is either bound in +width and unbound in depth or viceversa, its weights can then be set to represent virtually any function (i.e. a wide +variety of functions families). +The training problem thus consists in building networks with sets of weights so to instantiate or approximate the func- +tion that would solve the assigned task, or that represents the input-output relation. This search is not trivial: it can +be framed as the optimization problem for a functional over the ANN weights. Such functional, typically called "loss +function", associates the "errors" made on the training data to the neural net parameters (its weights), acting as a total +performance score. Approaching local minima of the loss function and improving the network performance on the +training data is the prerequisite to generalize on real world and unseen data. +DL is concerned with the use of deep ANNs, namely characterized by depth, stacking several intermediate, (hidden) +layers between input and output units. +As mentioned above, other dimensions being equal, depth increases the representational power of ANNs and, more +specifically, aims at modeling complicated functions as meaningful compositions of simpler ones. +As with their biological counterparts [4], depth is supposed to manage hierarchies of features from larger input por- +tions, capturing characteristics often inherent to real world objects and effective in modeling actual data. +Overall, depth is one of the key features that allowed to overcome historical limits [5] of simpler ANNs such as the +Perceptron. At the same time, depth comes with numerical and methodological hardships in models training. +Part of the difficulties arise as the search space for the optimal set of parameters grows considerably with the number +of layers (and their width as well). +Other issues are strictly numerical, since the training algorithms include long computation chains that may affect the +stability of training and learning. +2 + +Hence, new or rediscovered ideas in training protocols and mathematical optimization (e.g. applying the "backpropa- +gation of errors" algorithm to neural nets [6]) played an important role through times when the scientific interest and +hopes in ANNs faded (so called "AI winters"), paving the way for later advancement. +The main drivers for the latest success of deep neural networks are of varied nature, and can be schematised as techni- +cal and human related factors. +On a technical side DL has profited from [7]: +• the datafication of the world, i.e. the growing availability of (Big) data +• the diffusion of Graphical Processing Units (GPUs) as hardware tools. +To outperform classic machine learning models, deep neural networks often require larger quantities of data samples. +Such data hunger and high parameters count contribute to the high requirements of deep models in terms of memory, +number of operations and computation time. Training models with highly parallelized and smartly scheduled compu- +tations gained momentum thanks to GPUs. +In 2012 a milestone exemplified both the above technical aspects, when AlexNet [8], a deep Convolutional Neural +Network (CNN) based on ideas from Fukushima [4] and LeCun [9] - [10], won the ImageNet Large Scale Visual +Recognition Challenge after being trained using two GPUs [11]. Since then, Deep Learning has brought new out- +standing results in various tasks and domains, processing different data types. Deep networks can nowadays work on +image, video, audio, text, and speech data, time series and sequences, graphs, and more; the main tasks consist in +classification, prediction, or estimating the probability density of data distributions, with the possibility of modifying, +completing the input or even generating new instances. +On a more sociological side, the drivers of Deep Learning success can be related to the synergy of big tech companies, +advanced research centers, and developer communities [12]. Investments of economical and scientific resources in +relatively independent, collective projects, such as open-source libraries, frameworks, and APIs (Application Program- +ming Interfaces), have offered varied tools adapted to multiple specific situations and objectives, exploiting horizontal +organization [13] and mixing top-down and bottom-up approaches. It is difficult to imagine a rapid rise of successful +endeavors, without both active communities and the technical means to incorporate and manage lower-level aspects. +In fact, applying Deep Learning to a relevant problem in any research field requires, in addition to specific domain +knowledge, a vast background of statistical, mathematical, and programming notions and skills. The tools that support +scientists and engineers in focusing on their main tasks encompass the languages to express numerical operations on +GPUs, such as CUDA [14] and cuDNN [15] by NVIDIA, as well as the frameworks to design models, like Tensor- +Flow [16] and Keras [17] by Google, and PyTorch by Meta [18], or the supporting strategies to build data pipelines. +Many Deep Learning achievements are relevant to biomedical and clinical research, and the above presented tools +have enabled explorations of the capabilities of deep neural networks with neuroscience and biomedical data. +A fuller exploitation and routinely employment of modern algorithms are yet to come, both in research and clinical +practice. This process would accelerate by popularizing, democratizing, and jointly developing models, improving +their usability, and expanding their environments, i.e. by wrapping solutions into libraries and shared frameworks. +2.2 +Neuroscience +As per the Nature journal, «Neuroscience is a multidisciplinary science that is concerned with the study of the structure +and function of the nervous system. It encompasses the evolution, development, cellular and molecular biology, +physiology, anatomy and pharmacology of the nervous system, as well as computational, behavioural and cognitive +neuroscience» [19]. +Expanding, neuroscience investigates: +• the evolutionary and individual development of the nervous system; +• the cellular and molecular biology that characterizes neurons and glial cells; +• the physiology of living organisms and the role of the nervous system in the homeostatic function; +• the anatomy, i.e. the identification and description of the system’s structures; +• pharmacology, i.e. the effect of chemicals of external origin on the nervous system, their interactions with +endogenous molecules; +• the computational features of the brain and nerves, how information is processed, which mathematical and +physical models best predict and approximate the behaviour of neurons; +• cognition, the mental processes at the intersection of psychology and computational neuroscience; +• behaviour as a phenomenon rooted in genetics, development, mental states, and so forth. +3 + +The techniques to access tissues and structures of the nervous system are often shared by disciplines focused on other +physiological systems, and some of these processes have been computer aided for long. +Moreover, nerve cells have distinctive electromagnetic properties and their activity directly and indirectly generates +detectable signals, adding physical and technical specificity to Neuroscience. +Overall, neuroscience research is profoundly multi-modal. Data are managed and processed inside a model depending +on their type and format. The most prominent categories of data involved in neuroscience research comprise 2,3-D +images or video on the one side, and sequences or signals on the other. Still it is important to acknowledge the differ- +ent phenomena, autonomous or provoked by the measurement apparatus, underlying data generation and acquisition. +Bioimages may be produced from: +• Magnetic Resonance Imaging (MRI) +• X-rays +• Tomography with different penetrating waves +• Histopathology microscopy +• Fundus photography (retinal images) +and more. +Neuroscience sequences may come from: +• Electromiography (EMG) +• Electroencephalography (EEG) +• Natural language, text records +• Genetic sequencing +• Eye-tracking +and more. +Adding to the above, other data types are common in neuroscience, e.g. tabular data, text that may come from +medical records written by physicians for diagnostic purposes, test scores, inspections of cognitive and sensorimotor +functions, as the National Institute of Health (NIH) Stroke Scale test scores [20], and more broadly clinical reports +from anamneses or surveys. +2.3 +Neuroinformatics +Neuroscience is evolving into a data-centric discipline. Modern research heavily depends on human researchers as +well as machine agents to store, manage and process computerized data from the experimental apparatus to the end +stage. +Before delving in the specifics of artificial neural networks applied to the study of biological neural systems, it is +useful to outline the broader concepts of Neuroinformatics, regarding data and coding, especially in the light of open +culture. +According to the International Neuroinformatics Coordinating Facility (INCF), «Neuroinformatics is a research field +devoted to the development of neuroscience data and knowledge bases together with computational models and ana- +lytical tools for sharing, integration, and analysis of experimental data and advancement of theories about the nervous +system function. In the INCF context, neuroinformatics refers to scientific information about primary experimental +data, ontology, metadata, analytical tools, and computational models of the nervous system. The primary data includes +experiments and experimental conditions concerning the genomic, molecular, structural, cellular, networks, systems +and behavioural level, in all species and preparations in both the normal and disordered states» [21]. Given the rele- +vance of Neuroinformatics to Neuroscience, supporting open and reproducible science implies and requires attention +to standards and best practices regarding open data and code. +The INCF itself is an independent organization devoted to validate and promote such standards and practices, inter- +acting with the research communities [22] and aiming at the "FAIR principles for scientific data management and +stewardship" [23]. +FAIR principles consist in: +• being Findable, registered and indexed, searchable, richly described in metadata; +• being Accessible, through open, free, universally implementable protocols; +• being Interoperable, with appropriate standards for metadata in the context of knowledge representation; +4 + +• being Reusable, clearly licensed, well described, relevant to a domain and meeting community standards. +Among free and open resources, several software and organized packages integrating pre-processing and data analysis +workflows for neuroimaging and signal processing became the reference for worldwide researchers in Neuroscience. +Such tools allow to perform scientific research in neuroscience easily in solid and repeatable ways. It can be useful to +mention, for neuroimaging, Freesurfer1 [24] and FSL2 [25] that are standalone softwares, and the MATLAB-connected +SPM3 [26]. In the domain of signal processing, examples are EEGLAB4 [27], Brainstorm5 [28], PaWFE6 [29], all +MATLAB related yet free and open, and MNE7 [30], that runs on Python. Regarding applications for neurorobotics +and Brain Computer Interfaces (BCIs), a recent opensource platform can be found in ROS-neuro8 [31]. +The interested readers can find lists of open resources for computational neuroscience (including code, data, mod- +els, repositories, textbooks, analysis, simulation and management software) at Open Computational Neuroscience +Resource 9 (by Austin Soplata), and at Open Neuroscience 10. Additional software resources oriented to Neuroinfor- +matics in general, but not necessarily open, can also be found as indexed at "COMPUTATIONAL NEUROSCIENCE +on the Web" 11 (by Jim Perlewitz). +2.4 +Bringing Deep Learning to the Neurosciences +The Deep Learning community is accustomed to open science, as many datasets, models, programming frameworks +and scientific outcomes are publicly released by both academia and companies continuously. However, while Deep +Learning can openly provide state-of-the-art models to old and new problems in Neuroscience, theoretical understand- +ing, formalization and standardisation are often yet to be achieved, which may prevent adoption in other research +endeavors. From a technical standpoint, deep networks are a viable tool for many tasks involving data from the brain +sciences. Image classification has arguably been the task in which deep neural networks have had the highest mo- +mentum, in terms of pushing the state of the art forward. This translates now in a rich taxonomy of architectures +and pre-trained models that consistently maintain interesting performances in pattern recognition, across a number of +image domains. +Pattern recognition is indeed central for diagnostic purposes, in the form of classification of images with pathological +features (e.g. types of brain tumors or meningiomas), segmentation of structures (such as the brain, brain tumors +or stroke lesions), classification of signals (e.g. classification of electromyography or electro encephalography data), +as well as for action recognition in Human-Computer Interfaces (HCIs). The initiatives BRain Tumor Segmentation +(BRATS) Challenge12 [32], Ischemic Stroke LEsion Segmentation (ISLES) Challenge13 [33]- [34], and Ninapro14 [35] +are examples of data releases for which above-mentioned tools proved effective. +There are models learning image-to-image functions, capable of enhancing data, preprocessing it, correcting artifacts +and aberrations, allowing smart compression as well as super-resolution, and even expressing cross-modal transforma- +tions between different acquisition apparatus. +In the related tasks of object tracking, action recognition and pose estimation, research results from the automotive +sector or crowd analysis have inspired solutions for behavioural neuroscience, especially in animal behavioral studies. +When dealing with sequences, deep networks success in Computer Vision has inspired CNN-based approaches to EEG +and EMG studies [36] - [37], either with or without relying on 2D data, given that mathematical convolution has a 1D +version, and 1D signals have 2D spectra. Other architectures more directly instantiate temporal and sequential aspects, +e.g. Recurrent Neural Networks (RNNs) such as the Long Short Term Memory (LSTM) [38] and Gated Recurrent +Units (GRUs) [39], and they too can be applied to sequence problems and sub-tasks in neuroscience, such as decoding +time-dependent brain signals. +Although deep neural network do not explicitly model the nervous system, they are inspired by biological knowledge +and mimic some aspects of biological computation and dynamical systems. This has inspired new comparative studies, +1https://surfer.nmr.mgh.harvard.edu/ +2https://fsl.fmrib.ox.ac.uk/fsl/fslwiki +3https://www.fil.ion.ucl.ac.uk/spm/ +4https://sccn.ucsd.edu/eeglab/index.php +5https://neuroimage.usc.edu/brainstorm/Introduction +6http://ninapro.hevs.ch/node/229 +7https://mne.tools/stable/index.html +8https://github.com/rosneuro +9https://github.com/asoplata/open-computational-neuroscience-resources +10https://open-neuroscience.com/ +11https://compneuroweb.com/sftwr.html +12https://www.med.upenn.edu/cbica/brats/ +13https://www.isles-challenge.org/ +14http://ninaweb.hevs.ch/node/7 +5 + +and analogy approaches to learning and perception, in a unique way among machine learning algorithms [40]. +Many neuroinformatic studies demonstrate how novel deep learning concepts and methods apply to neurological +data [12]. However, they often showcase new further achievements in performance metrics that do not translate di- +rectly to new accepted neuroscience discoveries or clinical best practices. +Such results are very often published together with open code repositories, allowing reproducibility, yet they may not +be explicitly organized for widespread routinely adoption in domains different from machine learning. Algorithms are +usually written in open programming languages like Python [41], R [42], Julia [43], and deep learning design frame- +works such as TensorFlow, PyTorch or Flux [44]. Still, they are more inspiring to the experienced machine learning +researcher, rather than practically helpful to end-users such as neuroscientists. +In fact, to successfully build a deep learning application from scratch, a vast knowledge is needed in the data science +aspect of the task and in coding , as much as in the theoretical and experimental foundations and frontiers of the +application domain, here being Neuroscience. +For the above reasons, the open source and open science domains are promising frames for common development +and testing of relevant solutions for Neuroscience, as they provide an active flow of ideas and robust diversification, +avoiding "reinvention of the wheel", harmful redundancies or starting from completely blank states. +As a contribution in clarifying the current situation and reducing the workload for researchers, this work collects and +analyzes several open libraries that implement and facilitate Deep Learning application in Neuroscience, with the aim +of allowing worldwide scientists to identify the most suitable options for their inquiries and clinical tasks. +3 +Methods +The large corpus of available open code makes useful to specify what qualifies as a coding library or a framework, +rather than as a model accompanied by utilities, for the present scope. +In programming, a library is a collection of pre-coded functions and object definitions, often relying on one another, +and written to optimize programming for custom tasks. The functions are considered useful and unmodified across +multiple unrelated programs and tasks. The main program at hand calls the library, in the control flow specified by the +end-users. +A framework is a higher level concept, akin to the library, but typically with a pre-designed control flows in which +custom code from the end-users is inserted. +For instance, a repository that simply collects the functions that define and instantiate a deep model would not be +considered a library. On the other hand, collections of notebooks that allow to train, retrain and test models with +several architectures, while possibly taking care also of data pre-processing and preparation, would be considered +libraries (and frameworks) for the present scopes. +The explicit definition of the authors, their aims and their +maintainance of the library is relevant as well, in determining if a repository would be considered a library, toolkit, +toolbox, or other. +For the sake of the review, several resources were queried or scanned. Google Scholar was queried with: +• "deep learning library" OR "deep learning toolbox" OR "deep learning package" -"MATLAB deep learning +toolbox" -"deep learning toolbox MATLAB" +preserving the top 100 search results, ordered for relevance by the engine algorithm. On PubMed the queries were: +• opensource (deep learning) AND (toolbox OR toolkit OR library); +• (EEG OR EMG OR MRI OR (brain (X-ray OR CT OR PT))) (deep learning) AND (toolbox OR toolkit OR +library). +Moreover, the site https://open-neuroscience.com/ was scanned specifically for "deep learning" mentions, and +relevant papers cited or automatically suggested throughout the query process were considered for evaluation, as well +as the platform of the Journal of Open Source Software at https://joss.theoj.org/. +The collected libraries were organized according to the principal aim, in the form of data type processed, or the +supporting function in the workflow, thus dividing: +1. libraries for sequence data (e.g. EMG, EEG) +2. libraries for image data (including scalar volumes, 4-dimensional data as in fMRI, video) +3. libraries and frameworks to support model building, evaluation, data ingestion +In each category, a set of three tables present separately the results related to the following libraries characteristics: +6 + +1. domain of application +2. model engineering +3. technology and sources +The domain of application comprises the Neuroscience area, the Data types handled, the provision of Datasets, and +the machine learning Task to which the library is dedicated. +The model engineering tables include informations on the architecture of DL Models manageable in the library, the +DL framework and Programming language main dependencies, and the possibility of Customization for the model +structure or training parameters. +Technology and sources refer to the type of Interface available for a library, whether it works Online//Offline, specif- +ically with real-time data or logged data. Maintenance refers to the ongoing activity of releasing features, solving +issues and bugs or offering support through channels, Source specifies where code files and instructions are made +available. +4 +Results: Deep Learning Libraries +The analysis of the literature allowed to select a total of 48 publications describing libraries that implement or em- +power deep learning applications for neuroscience. Despite open source and effectiveness, several publications did not +provide an ecosystem of reusable functions. Proofs of concept and single-shot experiments were discarded. +4.1 +Libraries for sequence data +Libraries and frameworks for sequence data are shown in Tables 1 (domains of application), 2 (models characteris- +tics), 3 (technologies and sources). The majority of process EEG sygnals, which are among the most common types +of sequential data in Neuroscience research. A common objective is deducing the activity or state of the subject, +based on temporal or spectral (2D) patterns. Deep Learning is capable of bypassing some of the preprocessing steps +often required by other common statistical and engineering techniques, and it comprises both 1D and 2D approaches, +through MLPs, CNNs or RNNs architectures. BioPyC is an example of such scenario. It offers the possibility to +train a pre-set CNN architecture as well as loading and training a custom model. Moreover, It can process different +types of sequence data, making it very versatile and applicable/ suitable/usable in/for different neuroscience area. +Another example of sequence-oriented library is gumpy, whose intended area of application is that of Brain Computer +Interfaces (BCIs), where decoding a signal is the first step towards communication and interaction with a computer +or robotic system. Given the setting, gumpy allows working with EEG or EMG data and suits them with specific +defaults, e.g. 1-D CNNs, or LSTMs. +Notable mentions in the sequence category are the library Traja and the VARDNN toolbox, as they depart from +the common scenarios of previous examples. Traja stands out as an example of less usual sequential data, namely +trajectory data (sequences of coordinates in 2 or 3 dimensions, through time). Moreover, in Traja sequences are +modeled and analyzed employing the advanced architectures of Variational AutoEncoders (VAEs) and Generative +Adversarial Networks (GANs), usually encountered in image tasks. With different theoretical backgrounds, both +architectures allow simulation and characterization of data through their statistical properties. The VARDNN toolbox +allows analyses on BOLD signals, in the established domain of functional Magnetic Resonance Imaging (fMRI), but +uses a unique approach to autoregressive processes mixed with deep neural networks, allowing to perform causal +analysis and to study functional connections between brain regions through their patterns of activity in time. +7 + +Name +Neuroscience +area +Data type +Datasets +Task +BioPyC [45] +General +Sequences (EEG, miscellaneous) +No +Classification +braindecode [46] +General +Sequences (EEG, MEG) +External +Classification +DeLINEATE [47] +General +Images, sequences +External +Classification +EEG-DL [48] +BCI +Sequences (EEG) +No +Classification +gumpy [49] +BCI +Sequences (EEG, EMG) +No +Classification +DeepEEG +Electrophysiology +Sequences (EEG) +No +Classification +ExBrainable [50] +Electrophysiology +Sequences (EEG) +External +Classification, XAI +Traja [51] +Behavioural +neuro- +science +Sequences (Trajectory coordinates over time) +No +Prediction, Classification, Synthesis +VARDNN toolbox [52] toolbox +Connectomics +(Functional +Connectiv- +ity) +Sequences (BOLD signal) +No +Time series causal analysis +Table 1: Domains of applications for the libraries and frameworks processing sequence data +8 + +Name +Models +DL framework +Customization +Programming language +BioPyC +1-D CNN +Lasagne +Yes (weights, model) +Python +braindecode +1-D CNN +PyTorch +Yes (weights, model) +Python +DeLINEATE +CNN +Keras, TensorFlow +Yes (weights, model) +Python +EEG-DL +Miscellaneous +TensorFlow +Yes (weights, model) +Python, MATLAB +gumpy +CNN, LSTM +Keras, Theano +Yes (weights, model) +Python +DeepEEG +MLP, 1,2,3-D CNN, LSTM +Keras, TensorFlow +Yes (weights) +Python +ExBrainable +CNN +PyTorch +Yes (weights) +Python +Traja +LSTM, VAE, GAN +PyTorch +Yes (weights, model) +Python +VARDNN toolbox +Vector Auto-Regressive DNN +Deep Learning Toolbox (MATLAB) +Yes (weights) +MATLAB +Table 2: Model engineering specifications for the libraries and frameworks processing sequence data +9 + +Name +Interface +Online/Offline +Maintenance +Source +BioPyC +Jupyter Notebooks +Offline +Active +gitlab.inria.fr/biopyc/BioPyC/ +braindecode +None +Offline +Active +github.com/braindecode/braindecode +DeLINEATE +GUI, Colab Notebooks +Offline +Active +bitbucket.org/delineate/delineate +EEG-DL +None +Offline +Active +github.com/SuperBruceJia/EEG-DL +gumpy +None +Online, Offline +Inactive +github.com/gumpy-bci +DeepEEG +Colab Notebooks +Offline +Inactive +github.com/kylemath/DeepEEG +ExBrainable +GUI +Offline +Active +github.com/CECNL/ExBrainable +Traja +None +Offline +Active +github.com/traja-team/traja +VARDNN toolbox +None +Offline +Active +github.com/takuto-okuno-riken/vardnn +Table 3: Technological aspects and code sources for the libraries and frameworks processing sequence data +10 + +4.2 +Libraries for image data +Libraries and frameworks for image data are shown in Tables 4 (domains of application), 5 (models characteristics),6 +(technologies and sources). Computer vision and 2D image processing are arguably the fields in which DL has +achieved the most impressive and state-of-art defining results, often inspiring and translating breakthroughs in other +domanis. Classification and segmentation (i.e. the separation of parts of the image based on their classes) are the +most common tasks addressed by the image processing libraries. Magnetic resonance is the primary source of data; +however, various deep learning libraries are built microscopic and eye-tracking data as well. Most of the libraries +collected in our analysis take advantage of classical CNN architectures for classification, Convolutional AutoEncoders +(CAEs) for segmentation, and GANs for synthesis. It is common to employ transfer learning to lessen the compu- +tational and memory burden during the training phase, and take advantage of pre-trained models. Transfer learning +consists in initializing models with parameters learnt on usually larger data sets, possibly from different domains and +tasks, with varying amounts of further training in the target domain. The best such examples are pose-estimation +libraries extending the DeepLabCut system, arguably the most relevant project on the topic. DeepLabCut is an +interactive framework for labelling, training, testing and refining models, that originally exploits the weights learned +from ResNets (or newer architectures) on the ImageNet data. The results match human annotation using quite few +training samples, holding for many (human and non-human) animals, and settings. The documentation and demon- +strative notebooks and tools offered by the Mathis Lab allow different levels of understanding and customization of +the process, with high levels of robustness. Among the considered libraries, two set apart from the majority given +the type of tasks they perform: GaNDLF addresses eXplainable AI (XAI), i.e. Artificial Intelligence whose deci- +sions and outputs can be understood by humans through more transparent mental models; ANTsX performs both the +co-registration step and super-resolution as a quality enhancing step for neuroimages, with the former being usually +performed by traditional algorithms. GaNDLF sets its goal as the provision of deep learning resources in different +layers of abstraction, allowing medical researchers with virtually no ML knowledge to perform robust experiments +with models trained on carefully split data, with augmentations and preprocessing, under standardized protocols that +can easily integrate interpretability tools such as Grad-CAM [53] and attention maps, which highlight the parts of +an image according to how they influenced a model outcome. The ANTsX ecosystem is of similar wide scope, and +is intended to build workflows on quantitative biology and medical imaging data, both in Python and R languages. +Packages from the same ecosystem perform registration of brain structures (by classical methods) as well as brain +extraction by deep networks. +11 + +Name +Neuroscience +area +Data type +Datasets +Task +AxonDeepSeg [54] +Microbiology, +Histology +Img (SEM, TEM) +External +Segm. +DeepCINAC [55] +Electrophys. +Vid (2-photon calcium) +No +Class. +DeepLabCut [56] +Behavioral +neuroscience +Vid +No +Pose est. +DeepNeuro [57] +Neuroimaging +Img (fMRI, misc.) +No +Class., Segm., Synthesis +DeepVOG [58] +Oculography +Img, Vid +Demo +Segm. +DeLINEATE [47] +General +Img, sequences +External +Class. +DNNBrain [59] +Brain +map- +ping +Img +No +Class. +ivadomed [60] +Neuroimaging +Img (2D, 3D) +No +Class., Segm. +MEYE [61] +Oculography +Img, Vid +Yes +Segm. +Allen Cell Structure Segmenter [62] +Microbiology, +Histology +Img (3D-fluor. microscopy) +No +Segm. +VesicleSeg [63] +Microbiology, +Histology +Img (EM) +No +Segm. +CDeep3M2 [64] +Microbiology, +Histology +Img (misc. microscopy) +Yes +Segm. +CASCADE [65] +Electrophys. +Vid (2-photon calcium), Seq +Yes +Event detection +ScLimibic [66] +Neuroimaging +Img (MRI) +External +Segm. +ALMA [67] +Behavioral +neuroscience +Vid +External +Pose est., Class. +fetal-code [68] +Neuroimaging +Img (rs-fMRI) +External +Segm. +ClinicaDL [69] +Neuroimaging +Img (MRI, PET) +External +Class., Segm. +DeepNeuron [70] +Microbiology, +Histology +Img (confocal microscopy) +No +Obj. detect., Segm. +GaNDLF [71] +Medical +Imaging +Img (2D, 3D) +External +Segm., Regression, XAI +MesoNet [72] +Neuroimaging +Img (fluoresc. microscopy) +External +Segm., Registration +MARS, BENTO [73] +Behavioral +neuroscience +Vid +Yes +Pose est., Class., Action rec., Tag +NiftyNet [74] +Medical +Imaging +Img (MRI, CT) +No +Class., Segm., Synth. +ANTsX [75] (ANTsPyNet, ANTsRNet) +Neuroimaging +Img (MRI) +No +Classificastion, Segm., Registr., Super-res. +MARS, BENTO [73] +Behavioral +neuroscience +Vid +Yes +Pose est., Class., Action rec., Tag +Visual Fields Analysis [76] +Eye tracking, +Behavioral +neuroscience +Vid +No +Pose est., Class. +Table 4: Domains of applications for the libraries and frameworks processing image data +12 + +Name +Models +DL framework +Customization +Programming language +AxonDeepSeg +CAE +TensorFlow +Yes (weights) +Python +DeepCINAC +DeepCINAC +(CNN+LSTM) +Keras, TensorFlow +Yes (weights) +Python +DeepLabCut +CNN +TensorFlow +Yes (weights) +Python +DeepNeuro +CNN, CAE, GAN +Keras, TensorFlow +Yes (weights, model) +Python +DeepVOG +CAE +TensorFlow +No +Python +DeLINEATE +CNN +Keras, TensorFlow +Yes (weights, model) +Python +DNNBrain +CNN +PyTorch +Yes (model) +Python +ivadomed +2,3-D CNN, CAE +PyTorch +Yes (weights, model) +Python +MEYE +CAE, CNN +TensorFlow +Yes (model) +Python +Allen Cell Structure Segmenter +CAE +PyTorch +No +Python +VesicleSeg +CNN +PyTorch +No +Python +CDeep3M2 +CAE +TensorFlow +Yes (weights) +Python +CASCADE +1-D CNN +TensorFlow +Yes (weights) +Python +ScLimibic +3-D CAE +neurite, TensorFlow +No +Python +ALMA +CNN +Unspecified +No +Python +fetal-code +2-D CNN +TensorFlow +No +Python +ClinicaDL +CNN, CAE +PyTorch +Yes +Python +DeepNeuron +CNN +Unspecified +No +C++ +GaNDLF +CNN, CAE +PyTorch +Yes +Python +MesoNet +CNN, CAE +Keras, TensorFlow +No +Python +NiftyNet +CNN +TensorFlow +Yes +Python +ANTsX (ANTsPyNet, ANTsRNet) +CNN, CAE, GAN +Keras, TensorFlow +Yes +Python, R, C++ +MARS, BENTO +CNN +TensorFlow +Yes (weights) +Python +Visual Fields Analysis +DeepLabCut +TensorFlow, DeepLabCut +Yes (weights) +Python +Table 5: Model engineering specifications for the libraries and frameworks processing image data +13 + +Name +Interface +Online/Offline +Maintenance +Source +AxonDeepSeg +Jupyter +Note- +books +Offline +Active +github.com/axondeepseg/axondeepseg +DeepCINAC +GUI, +Colab +Notebooks +Offline +Active +gitlab.com/cossartlab/deepcinac +DeepLabCut +GUI, +Colab +Notebooks +Offline +Active +github.com/DeepLabCut/DeepLabCut +DeepNeuro +None +Offline +Active +github.com/QTIM-Lab/DeepNeuro +DeepVOG +None +Offline +Inactive +github.com/pydsgz/DeepVOG +DeLINEATE +GUI, +Colab +Notebooks +Offline +Active +bitbucket.org/delineate/delineate +DNNBrain +None +Offline +Active +github.com/BNUCNL/dnnbrain +ivadomed +None +Offline +Active +github.com/ivadomed/ivadomed +MEYE +Web app +Online, Offline +Active +pupillometry.it +Allen Cell Structure Segmenter +GUI, +Jupyter +Notebooks +Offline +Active +github.com/AllenCell/aics-ml-segmentation +VesicleSeg +GUI +Offline +Active +github.com/Imbrosci/synaptic-vesicles-detection +CDeep3M2 +GUI, +Colab +Notebooks +Offline +Active +github.com/CRBS/cdeep3m2 +CASCADE +GUI, +Colab +Notebooks +Offline +Active +github.com/HelmchenLabSoftware/Cascade +ScLimibic +Unspecified +Offline +Active +surfer.nmr.mgh.harvard.edu/fswiki/ScLimbic +ALMA +GUI +Offline +Active +github.com/sollan/alma +fetal-code +GUI, +Colab +Notebooks +Offline +Active +github.com/saigerutherford/fetal-code +ClinicaDL +GUI, +Colab +Notebooks +Offline +Active +github.com/aramis-lab/clinicadl +DeepNeuron +GUI +Online, Offline +Inactive +github.com/Vaa3D/Vaa3D_Data/releases/tag/1.0 +GaNDLF +GUI +Offline +Active +github.com/CBICA/GaNDLF +MesoNet +GUI, +Colab +Notebooks +Offline +Active +osf.io/svztu +NiftyNet +None +Offline +Inactive +github.com/NifTK/NiftyNet +ANTsX (ANTsPyNet, ANTsRNet) +None +Offline +Active +github.com/ANTsX +MARS, BENTO +GUI, +MATLAB +GUI, +Jupyter +Notebooks +Offline +Active +github.com/neuroethology +Visual Fields Analysis +GUI +Offline +Active +github.com/mathjoss/VisualFieldsAnalysis +Table 6: Technological aspects and code sources for the libraries and frameworks processing image data +14 + +4.3 +Libraries targeting data types different from sequences or images and general applications +Libraries and frameworks for sequence data are shown in Tables 7 (domains of application), 8 (models characteris- +tics), 9 (technologies and sources). In this category fall libraries and projects with either varying input data type, or +other than sequence and image data analysis; other libraries target computational platforms, higher hierarchy frame- +works, or supporting functions for deep learning like specific preprocessing and augmentations. NeuroCAAS is an +ambitious project that both standardizes experimental schedules, analyses and offers computational resources on the +cloud. The platform lifts the burden of configuring and deploying data analysis tool, guaranteeing also replicability +and readily available usage of pre-made pipelines, with high efficiency. MONAI is a project that brings deep learning +tools to many health and biology problems, and is a commonly used framework for the 3D variations of UNet [77] +lately dominating the yearly BraTS challenge [32] (see at http://braintumorsegmentation.org/). The paradigm +builds on PyTorch and aims at unifying healthcare AI practices throughout both academia and enterprise research, not +only in the model development but also in the creation of shared annotated datasets. Lastly, it focuses on deployment +and work in real world clinical production, settling as a strong candidate for being the standard solution in the do- +main. Predify and THINGvision are two libraries that bridge deep learning research and computational neuroscience. +The former allows to include an implementation of a «predictive coding mechanism» (as hypothesized in [78]) into +virtually any pre-built architectures, evaluating its impact on performance. The latter offers a single environment for +Representational Similarity Analysis, i.e. the study of the encodings of biological and artificial neural networks that +process visual data. +15 + +Name +Neuroscience area +Data type +Datasets +Task +NeuroCAAS [79] +Virtually all +Virtually all +External availability +Virtually all +MONAI [80] +Virtually all +Virtually all +External availability +Virtually all +Predify [81] +Computational +Neuro- +science +Images, Virtually all +No +Classification, Adversarial attacks, virtually all +THINGvision [82] +Computational +Neuro- +science +Images, Text +External availability +Classification +TorchIO [83] +Imaging +All images +No +Augmentation +Table 7: Domains of applications for the libraries and frameworks for special applications +16 + +Name +Models +DL framework +Customization +Programming language +NeuroCAAS +CNN +TensorFlow +Yes +Python +MONAI +Virtually All +PyTorch +Yes +Python +Predify +CNN, Virtually all +PyTorch +Yes +Python +THINGvision +CNN, RNN, Transformers +PyTorch, TensorFlow +No +Python +TorchIO +CNN +PyTorch +Yes +Python +Table 8: Model engineering specifications for the libraries and frameworks for special applications +17 + +Name +Interface +Online/Offline +Maintenance +Source +NeuroCAAS +GUI, Jupyter Notebooks +Offline +Active +github.com/cunningham-lab/neurocaas +MONAI +GUI, Colab Notebooks +Offline +Active +github.com/Project-MONAI/MONAI +Predify +Text UI (TOML) +Offline +Active +github.com/miladmozafari/predify +THINGvision +None +Offline +Active +github.com/ViCCo-Group/THINGSvision +TorchIO +GUI, Command line +Offline +Active +torchio.rtfd.io +Table 9: Technological aspects and code sources for the libraries and frameworks for special applications +18 + +5 +Discussion +The panorama of open-source libraries dedicated to deep learning applications in neuroscience is quite rich and diver- +sified. There is a corpus of organized packages that integrate preprocessing, training, testing and performance analyses +of deep neural networks for neurological research. Most of these projects are tuned to specific data modalities and +formats, but some libraries are quite versatile and customizabile, and there are projects that encompass quantitative +biology and medical analysis as a whole. There is a common tendency to develop GUIs, enhancing user-friendliness of +toolkits for non-programmers and researchers unacquainted with the command line interfaces, for example. Moreover, +for the many libraries developed in Python, the (Jupyter) Notebook format appears as a widespread tool both for tutori- +als, documentation and as an interface to cloud computational resources (e.g. Google Colab [84]). Apart from specific +papers and documentation, and outside of deep learning per se, it is important to make researchers and developers +aware of the main topics and initiatives in open culture and Neuroinformatics. For this reason, the interested reader +is invited to rely on competent institutions (e.g. INCF) and databases of open resources (e.g. open-neuroscience) +dedicated to Neuroscience. Among the possibly missing technologies, the queries employed did not retrieve results +in Natural Language Processing libraries dedicated to neuroscience, nor toolkits specifically employing Graph Neural +Networks (GNNs), although available in EEG-DL. NLP is actually fundamental in healthcare, since medical reports +often come in non standardized forms. Large language models, Named Entity Recognition (NER) systems and text +mining approaches in biomedical research exist [85], [86]. GNNs comprise recent architectures that are extremely +promising in a variety of fields [87], including biomedical research and particularly neuroscience [88], [89]. Even if +promising, their application is still less mature than that of computer vision models or time series analysis. +Considering the available software for imaging and signal processing in the domain of neuroscience, at this moment a +single alternative targeting the opportunities offered by modern deep learning seems to be missing. Overall, it seems +still unlikely to develop a common deep learning framework for Neuroscience as a separate whole, but the engineering +knowledge relevant and compressible into such framework would be common to other biomedical fields, and projects +such as MONAI are strong candidates toward this goal. Instead, it seems achievable to deliver models and functions +in a concerted way, restricted either to a sub-field or a data modality, based on the modularity of existent tools and the +organizing possibilities of project initiation and management of open culture. +6 +Conclusions +Although a large and growing number of repositories offer code to build specific models, as published in experimental +papers, these resources seldom aim to constitute proper libraries or frameworks for research or clinical practice. Both +deep learning and neuroscience gain much value even from sophisticated proofs of concept. In parallel, organized +packages are spreading and starting to provide and integrate pre-processing, training, testing and performance analyses +of deep neural networks for neurological and biomedical research. This paper has offered both an historical and a +technical context for the use of deep neural networks in Neuroinformatics, focusing on open-source tools that scientists +can comprehend and adapt to their necessities. At the same time, this work underlines the value of the open culture and +points to relevant institutions and platforms for neuroscientists. 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The +Lancet Digital Health, 1(5):e232–e242, September 2019. +25 + diff --git a/0dE4T4oBgHgl3EQfZgyj/content/tmp_files/load_file.txt b/0dE4T4oBgHgl3EQfZgyj/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a1b192a432095e8debce4367b41c6b3492ff3657 --- /dev/null +++ b/0dE4T4oBgHgl3EQfZgyj/content/tmp_files/load_file.txt @@ -0,0 +1,983 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf,len=982 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='05057v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='QM] 19 Dec 2022 AN OVERVIEW OF OPEN SOURCE DEEP LEARNING-BASED LIBRARIES FOR NEUROSCIENCE Louis Fabrice Tshimanga Department of Neuroscience (DNS) University of Padova louisfabrice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='tshimanga@unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='it Manfredo Atzori Department of Neuroscience (DNS), Padova Neuroscience Center (PNC) University of Padova Information Systems Institute University of Applied Sciences Western Switzerland (HES-SO Valais) manfredo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='atzori@unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='it Federico Del Pup Department of Neuroscience (DNS), Department of Information Engineering (DEI) University of Padova federico.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='delpup@studenti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='it Maurizio Corbetta Department of Neuroscience (DNS), Padova Neuroscience Center (PNC) University of Padova Department of Neurology Washington University School of Medicine maurizio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='corbetta@unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='it ABSTRACT In recent years, deep learning revolutionized machine learning and its applications, producing re- sults comparable to human experts in several domains, including neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Each year, hundreds of scientific publications present applications of deep neural networks for biomedical data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Due to the fast growth of the domain, it could be a complicated and extremely time-consuming task for worldwide researchers to have a clear perspective of the most recent and advanced software libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' This work contributes to clarify the current situation in the domain, outlining the most useful libraries that implement and facilitate deep learning application to neuroscience, allowing scientists to identify the most suitable options for their research or clinical projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' This paper summarizes the main developments in Deep Learning and their relevance to Neuroscience;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' it then reviews neuroinformatic toolboxes and libraries, collected from the literature and from specific hubs of software projects oriented to neuroscience research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The selected tools are presented in tables detailing key features grouped by domain of application (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' data type, neuroscience area, task), model engineering (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' programming language, model customization) and technological aspect (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' interface, code source).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The results show that, among a high number of available software tools, several libraries are standing out in terms of functionalities for neuroscience applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The aggregation and discussion of this information can help the neuroscience community to devolop their research projects more efficiently and quickly, both by means of readily available tools, and by knowing which modules may be improved, connected or added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Keywords Deep Learning · Neuroscience · Neuroinformatics · Open source 1 Introduction In the last decade, Deep Learning (DL) has taken over most classic approaches in Machine Learning (ML), Computer Vision, Natural Language Processing, showing an unprecedented versatility, and matching or surpassing the perfor- mances of human experts in narrow tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The recent growth of DL applications to several domains, including Neuroscience, consequently offers numerous open- source software opportunities for researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Mapping available resources can allow a faster and more precise exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Neuroscience is a diversified field on its own, as much for the objects and scales it focuses on, as for the types of data it relies on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The discipline is also historically tied to developments in electrical, electronic, and information technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Modern Neuroscience relies on computerization in many aspects of data generation, acquisition, and analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Statistical and Machine Learning techniques already empower many software packages, that have become de facto standards in sev- eral subfields of Neuroscience, such as Principal and Independent Component Analysis in Electroencephalography and Neuroimaging, to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Meanwhile, the rich and rapidly evolving taxonomy of Deep Neural Networks (DNNs) is becoming both an opportu- nity and hindrance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' On the one hand, currently open-source DL libraries allow an increasing number of applications and studies in Neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' On the other hand, the adoption of available methods is slowed down by a lack of stan- dards, reference frameworks and established workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Scientific communities whose primary focus or background is not in machine learning engineering may be left partially aside from the ongoing Artificial Intelligence (AI) gold rush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' For such reasons it is fundamental to overview open-source libraries and toolkits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Framing a panorama could help researchers in selecting ready-made tools and solutions when convenient, as well as in pointing out and filling in the blanks with new applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' This work would contribute to advancing the community’s possibilities, reducing the workload for researchers to exploit DL, allowing Neuroscience to benefit of its most recent advancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 2 Background 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='1 Deep Learning Deep Learning (DL) has contributed many best solutions to problems in its parent field, Machine Learning, thanks to theoretical and technological achievements that unlocked its intrinsic versatility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Machine Learning is the study of computer algorithms that tackle problems without complete access to predefined rules or analytical, closed-form solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The algorithms often require a training phase to adjust parameters and satisfy internal or external constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' of exactness, approximation or generality) on dedicated data for which solutions might be already known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Machine Learning comprises a wide array of statistical and mathematical methods, including Artificial Neural Net- works (ANNs), biologically inspired systems that connect inputs and outputs through simple computing units (neu- rons), which act as function approximators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Each unit implements a nonlinear function of the weighted sum of its inputs, thus the output of the whole ANN is a composite function, as formally intended in mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The networks of neurons are most often layered and "feed- forward", meaning that units from any layer only output results to units in subsequent layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The width of a layer refers to its neuron count, while the depth of a network refers to its layer count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The typical architecture instantiating the above characteristics is the MultiLayer Perceptron [1] (MLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Universal approximation theorems [2] [3] ensure that, whenever a nonlinear network as the MLP is either bound in width and unbound in depth or viceversa, its weights can then be set to represent virtually any function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' a wide variety of functions families).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The training problem thus consists in building networks with sets of weights so to instantiate or approximate the func- tion that would solve the assigned task, or that represents the input-output relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' This search is not trivial: it can be framed as the optimization problem for a functional over the ANN weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Such functional, typically called "loss function", associates the "errors" made on the training data to the neural net parameters (its weights), acting as a total performance score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Approaching local minima of the loss function and improving the network performance on the training data is the prerequisite to generalize on real world and unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' DL is concerned with the use of deep ANNs, namely characterized by depth, stacking several intermediate, (hidden) layers between input and output units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' As mentioned above, other dimensions being equal, depth increases the representational power of ANNs and, more specifically, aims at modeling complicated functions as meaningful compositions of simpler ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' As with their biological counterparts [4], depth is supposed to manage hierarchies of features from larger input por- tions, capturing characteristics often inherent to real world objects and effective in modeling actual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Overall, depth is one of the key features that allowed to overcome historical limits [5] of simpler ANNs such as the Perceptron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' At the same time, depth comes with numerical and methodological hardships in models training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Part of the difficulties arise as the search space for the optimal set of parameters grows considerably with the number of layers (and their width as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Other issues are strictly numerical, since the training algorithms include long computation chains that may affect the stability of training and learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 2 Hence, new or rediscovered ideas in training protocols and mathematical optimization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' applying the "backpropa- gation of errors" algorithm to neural nets [6]) played an important role through times when the scientific interest and hopes in ANNs faded (so called "AI winters"), paving the way for later advancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The main drivers for the latest success of deep neural networks are of varied nature, and can be schematised as techni- cal and human related factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' On a technical side DL has profited from [7]: the datafication of the world, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' the growing availability of (Big) data the diffusion of Graphical Processing Units (GPUs) as hardware tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' To outperform classic machine learning models, deep neural networks often require larger quantities of data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Such data hunger and high parameters count contribute to the high requirements of deep models in terms of memory, number of operations and computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Training models with highly parallelized and smartly scheduled compu- tations gained momentum thanks to GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' In 2012 a milestone exemplified both the above technical aspects, when AlexNet [8], a deep Convolutional Neural Network (CNN) based on ideas from Fukushima [4] and LeCun [9] - [10], won the ImageNet Large Scale Visual Recognition Challenge after being trained using two GPUs [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Since then, Deep Learning has brought new out- standing results in various tasks and domains, processing different data types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Deep networks can nowadays work on image, video, audio, text, and speech data, time series and sequences, graphs, and more;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' the main tasks consist in classification, prediction, or estimating the probability density of data distributions, with the possibility of modifying, completing the input or even generating new instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' On a more sociological side, the drivers of Deep Learning success can be related to the synergy of big tech companies, advanced research centers, and developer communities [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Investments of economical and scientific resources in relatively independent, collective projects, such as open-source libraries, frameworks, and APIs (Application Program- ming Interfaces), have offered varied tools adapted to multiple specific situations and objectives, exploiting horizontal organization [13] and mixing top-down and bottom-up approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' It is difficult to imagine a rapid rise of successful endeavors, without both active communities and the technical means to incorporate and manage lower-level aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' In fact, applying Deep Learning to a relevant problem in any research field requires, in addition to specific domain knowledge, a vast background of statistical, mathematical, and programming notions and skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The tools that support scientists and engineers in focusing on their main tasks encompass the languages to express numerical operations on GPUs, such as CUDA [14] and cuDNN [15] by NVIDIA, as well as the frameworks to design models, like Tensor- Flow [16] and Keras [17] by Google, and PyTorch by Meta [18], or the supporting strategies to build data pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Many Deep Learning achievements are relevant to biomedical and clinical research, and the above presented tools have enabled explorations of the capabilities of deep neural networks with neuroscience and biomedical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' A fuller exploitation and routinely employment of modern algorithms are yet to come, both in research and clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' This process would accelerate by popularizing, democratizing, and jointly developing models, improving their usability, and expanding their environments, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' by wrapping solutions into libraries and shared frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='2 Neuroscience As per the Nature journal, «Neuroscience is a multidisciplinary science that is concerned with the study of the structure and function of the nervous system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' It encompasses the evolution, development, cellular and molecular biology, physiology, anatomy and pharmacology of the nervous system, as well as computational, behavioural and cognitive neuroscience» [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Expanding, neuroscience investigates: the evolutionary and individual development of the nervous system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' the cellular and molecular biology that characterizes neurons and glial cells;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' the physiology of living organisms and the role of the nervous system in the homeostatic function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' the anatomy, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' the identification and description of the system’s structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' pharmacology, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' the effect of chemicals of external origin on the nervous system, their interactions with endogenous molecules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' the computational features of the brain and nerves, how information is processed, which mathematical and physical models best predict and approximate the behaviour of neurons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' cognition, the mental processes at the intersection of psychology and computational neuroscience;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' behaviour as a phenomenon rooted in genetics, development, mental states, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 3 The techniques to access tissues and structures of the nervous system are often shared by disciplines focused on other physiological systems, and some of these processes have been computer aided for long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Moreover, nerve cells have distinctive electromagnetic properties and their activity directly and indirectly generates detectable signals, adding physical and technical specificity to Neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Overall, neuroscience research is profoundly multi-modal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Data are managed and processed inside a model depending on their type and format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The most prominent categories of data involved in neuroscience research comprise 2,3-D images or video on the one side, and sequences or signals on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Still it is important to acknowledge the differ- ent phenomena, autonomous or provoked by the measurement apparatus, underlying data generation and acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Bioimages may be produced from: Magnetic Resonance Imaging (MRI) X-rays Tomography with different penetrating waves Histopathology microscopy Fundus photography (retinal images) and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Neuroscience sequences may come from: Electromiography (EMG) Electroencephalography (EEG) Natural language, text records Genetic sequencing Eye-tracking and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Adding to the above, other data types are common in neuroscience, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' tabular data, text that may come from medical records written by physicians for diagnostic purposes, test scores, inspections of cognitive and sensorimotor functions, as the National Institute of Health (NIH) Stroke Scale test scores [20], and more broadly clinical reports from anamneses or surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='3 Neuroinformatics Neuroscience is evolving into a data-centric discipline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Modern research heavily depends on human researchers as well as machine agents to store, manage and process computerized data from the experimental apparatus to the end stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Before delving in the specifics of artificial neural networks applied to the study of biological neural systems, it is useful to outline the broader concepts of Neuroinformatics, regarding data and coding, especially in the light of open culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' According to the International Neuroinformatics Coordinating Facility (INCF), «Neuroinformatics is a research field devoted to the development of neuroscience data and knowledge bases together with computational models and ana- lytical tools for sharing, integration, and analysis of experimental data and advancement of theories about the nervous system function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' In the INCF context, neuroinformatics refers to scientific information about primary experimental data, ontology, metadata, analytical tools, and computational models of the nervous system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The primary data includes experiments and experimental conditions concerning the genomic, molecular, structural, cellular, networks, systems and behavioural level, in all species and preparations in both the normal and disordered states» [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Given the rele- vance of Neuroinformatics to Neuroscience, supporting open and reproducible science implies and requires attention to standards and best practices regarding open data and code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The INCF itself is an independent organization devoted to validate and promote such standards and practices, inter- acting with the research communities [22] and aiming at the "FAIR principles for scientific data management and stewardship" [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' FAIR principles consist in: being Findable, registered and indexed, searchable, richly described in metadata;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' being Accessible, through open, free, universally implementable protocols;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' being Interoperable, with appropriate standards for metadata in the context of knowledge representation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 4 being Reusable, clearly licensed, well described, relevant to a domain and meeting community standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Among free and open resources, several software and organized packages integrating pre-processing and data analysis workflows for neuroimaging and signal processing became the reference for worldwide researchers in Neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Such tools allow to perform scientific research in neuroscience easily in solid and repeatable ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' It can be useful to mention, for neuroimaging, Freesurfer1 [24] and FSL2 [25] that are standalone softwares, and the MATLAB-connected SPM3 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' In the domain of signal processing, examples are EEGLAB4 [27], Brainstorm5 [28], PaWFE6 [29], all MATLAB related yet free and open, and MNE7 [30], that runs on Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Regarding applications for neurorobotics and Brain Computer Interfaces (BCIs), a recent opensource platform can be found in ROS-neuro8 [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The interested readers can find lists of open resources for computational neuroscience (including code, data, mod- els, repositories, textbooks, analysis, simulation and management software) at Open Computational Neuroscience Resource 9 (by Austin Soplata), and at Open Neuroscience 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Additional software resources oriented to Neuroinfor- matics in general, but not necessarily open, can also be found as indexed at "COMPUTATIONAL NEUROSCIENCE on the Web" 11 (by Jim Perlewitz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='4 Bringing Deep Learning to the Neurosciences The Deep Learning community is accustomed to open science, as many datasets, models, programming frameworks and scientific outcomes are publicly released by both academia and companies continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' However, while Deep Learning can openly provide state-of-the-art models to old and new problems in Neuroscience, theoretical understand- ing, formalization and standardisation are often yet to be achieved, which may prevent adoption in other research endeavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' From a technical standpoint, deep networks are a viable tool for many tasks involving data from the brain sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Image classification has arguably been the task in which deep neural networks have had the highest mo- mentum, in terms of pushing the state of the art forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' This translates now in a rich taxonomy of architectures and pre-trained models that consistently maintain interesting performances in pattern recognition, across a number of image domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Pattern recognition is indeed central for diagnostic purposes, in the form of classification of images with pathological features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' types of brain tumors or meningiomas), segmentation of structures (such as the brain, brain tumors or stroke lesions), classification of signals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' classification of electromyography or electro encephalography data), as well as for action recognition in Human-Computer Interfaces (HCIs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The initiatives BRain Tumor Segmentation (BRATS) Challenge12 [32], Ischemic Stroke LEsion Segmentation (ISLES) Challenge13 [33]- [34], and Ninapro14 [35] are examples of data releases for which above-mentioned tools proved effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' There are models learning image-to-image functions, capable of enhancing data, preprocessing it, correcting artifacts and aberrations, allowing smart compression as well as super-resolution, and even expressing cross-modal transforma- tions between different acquisition apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' In the related tasks of object tracking, action recognition and pose estimation, research results from the automotive sector or crowd analysis have inspired solutions for behavioural neuroscience, especially in animal behavioral studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' When dealing with sequences, deep networks success in Computer Vision has inspired CNN-based approaches to EEG and EMG studies [36] - [37], either with or without relying on 2D data, given that mathematical convolution has a 1D version, and 1D signals have 2D spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Other architectures more directly instantiate temporal and sequential aspects, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Recurrent Neural Networks (RNNs) such as the Long Short Term Memory (LSTM) [38] and Gated Recurrent Units (GRUs) [39], and they too can be applied to sequence problems and sub-tasks in neuroscience, such as decoding time-dependent brain signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Although deep neural network do not explicitly model the nervous system, they are inspired by biological knowledge and mimic some aspects of biological computation and dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' This has inspired new comparative studies, 1https://surfer.' metadata={'source': 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+page_content='hevs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='ch/node/7 5 and analogy approaches to learning and perception, in a unique way among machine learning algorithms [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Many neuroinformatic studies demonstrate how novel deep learning concepts and methods apply to neurological data [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' However, they often showcase new further achievements in performance metrics that do not translate di- rectly to new accepted neuroscience discoveries or clinical best practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Such results are very often published together with open code repositories, allowing reproducibility, yet they may not be explicitly organized for widespread routinely adoption in domains different from machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Algorithms are usually written in open programming languages like Python [41], R [42], Julia [43], and deep learning design frame- works such as TensorFlow, PyTorch or Flux [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Still, they are more inspiring to the experienced machine learning researcher, rather than practically helpful to end-users such as neuroscientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' In fact, to successfully build a deep learning application from scratch, a vast knowledge is needed in the data science aspect of the task and in coding , as much as in the theoretical and experimental foundations and frontiers of the application domain, here being Neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' For the above reasons, the open source and open science domains are promising frames for common development and testing of relevant solutions for Neuroscience, as they provide an active flow of ideas and robust diversification, avoiding "reinvention of the wheel", harmful redundancies or starting from completely blank states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' As a contribution in clarifying the current situation and reducing the workload for researchers, this work collects and analyzes several open libraries that implement and facilitate Deep Learning application in Neuroscience, with the aim of allowing worldwide scientists to identify the most suitable options for their inquiries and clinical tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 3 Methods The large corpus of available open code makes useful to specify what qualifies as a coding library or a framework, rather than as a model accompanied by utilities, for the present scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' In programming, a library is a collection of pre-coded functions and object definitions, often relying on one another, and written to optimize programming for custom tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The functions are considered useful and unmodified across multiple unrelated programs and tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The main program at hand calls the library, in the control flow specified by the end-users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' A framework is a higher level concept, akin to the library, but typically with a pre-designed control flows in which custom code from the end-users is inserted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' For instance, a repository that simply collects the functions that define and instantiate a deep model would not be considered a library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' On the other hand, collections of notebooks that allow to train, retrain and test models with several architectures, while possibly taking care also of data pre-processing and preparation, would be considered libraries (and frameworks) for the present scopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The explicit definition of the authors, their aims and their maintainance of the library is relevant as well, in determining if a repository would be considered a library, toolkit, toolbox, or other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' For the sake of the review, several resources were queried or scanned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Google Scholar was queried with: "deep learning library" OR "deep learning toolbox" OR "deep learning package" -"MATLAB deep learning toolbox" -"deep learning toolbox MATLAB" preserving the top 100 search results, ordered for relevance by the engine algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' On PubMed the queries were: opensource (deep learning) AND (toolbox OR toolkit OR library);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' (EEG OR EMG OR MRI OR (brain (X-ray OR CT OR PT))) (deep learning) AND (toolbox OR toolkit OR library).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Moreover, the site https://open-neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/ was scanned specifically for "deep learning" mentions, and relevant papers cited or automatically suggested throughout the query process were considered for evaluation, as well as the platform of the Journal of Open Source Software at https://joss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='theoj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The collected libraries were organized according to the principal aim, in the form of data type processed, or the supporting function in the workflow, thus dividing: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' libraries for sequence data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' EMG, EEG) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' libraries for image data (including scalar volumes, 4-dimensional data as in fMRI, video) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' libraries and frameworks to support model building, evaluation, data ingestion In each category, a set of three tables present separately the results related to the following libraries characteristics: 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' domain of application 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' model engineering 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' technology and sources The domain of application comprises the Neuroscience area, the Data types handled, the provision of Datasets, and the machine learning Task to which the library is dedicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The model engineering tables include informations on the architecture of DL Models manageable in the library, the DL framework and Programming language main dependencies, and the possibility of Customization for the model structure or training parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Technology and sources refer to the type of Interface available for a library, whether it works Online//Offline, specif- ically with real-time data or logged data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Maintenance refers to the ongoing activity of releasing features, solving issues and bugs or offering support through channels, Source specifies where code files and instructions are made available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 4 Results: Deep Learning Libraries The analysis of the literature allowed to select a total of 48 publications describing libraries that implement or em- power deep learning applications for neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Despite open source and effectiveness, several publications did not provide an ecosystem of reusable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Proofs of concept and single-shot experiments were discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='1 Libraries for sequence data Libraries and frameworks for sequence data are shown in Tables 1 (domains of application), 2 (models characteris- tics), 3 (technologies and sources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The majority of process EEG sygnals, which are among the most common types of sequential data in Neuroscience research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' A common objective is deducing the activity or state of the subject, based on temporal or spectral (2D) patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Deep Learning is capable of bypassing some of the preprocessing steps often required by other common statistical and engineering techniques, and it comprises both 1D and 2D approaches, through MLPs, CNNs or RNNs architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' BioPyC is an example of such scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' It offers the possibility to train a pre-set CNN architecture as well as loading and training a custom model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Moreover, It can process different types of sequence data, making it very versatile and applicable/ suitable/usable in/for different neuroscience area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Another example of sequence-oriented library is gumpy, whose intended area of application is that of Brain Computer Interfaces (BCIs), where decoding a signal is the first step towards communication and interaction with a computer or robotic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Given the setting, gumpy allows working with EEG or EMG data and suits them with specific defaults, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 1-D CNNs, or LSTMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Notable mentions in the sequence category are the library Traja and the VARDNN toolbox, as they depart from the common scenarios of previous examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Traja stands out as an example of less usual sequential data, namely trajectory data (sequences of coordinates in 2 or 3 dimensions, through time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Moreover, in Traja sequences are modeled and analyzed employing the advanced architectures of Variational AutoEncoders (VAEs) and Generative Adversarial Networks (GANs), usually encountered in image tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' With different theoretical backgrounds, both architectures allow simulation and characterization of data through their statistical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The VARDNN toolbox allows analyses on BOLD signals, in the established domain of functional Magnetic Resonance Imaging (fMRI), but uses a unique approach to autoregressive processes mixed with deep neural networks, allowing to perform causal analysis and to study functional connections between brain regions through their patterns of activity in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 7 Name Neuroscience area Data type Datasets Task BioPyC [45] General Sequences (EEG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' miscellaneous) No Classification braindecode [46] General Sequences (EEG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' MEG) External Classification DeLINEATE [47] General Images,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' sequences External Classification EEG-DL [48] BCI Sequences (EEG) No Classification gumpy [49] BCI Sequences (EEG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' EMG) No Classification DeepEEG Electrophysiology Sequences (EEG) No Classification ExBrainable [50] Electrophysiology Sequences (EEG) External Classification,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' XAI Traja [51] Behavioural neuro- science Sequences (Trajectory coordinates over time) No Prediction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Classification,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Synthesis VARDNN toolbox [52] toolbox Connectomics (Functional Connectiv- ity) Sequences (BOLD signal) No Time series causal analysis Table 1: Domains of applications for the libraries and frameworks processing sequence data 8 Name Models DL framework Customization Programming language BioPyC 1-D CNN Lasagne Yes (weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' model) Python braindecode 1-D CNN PyTorch Yes (weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' model) Python DeLINEATE CNN Keras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' TensorFlow Yes (weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' model) Python EEG-DL Miscellaneous TensorFlow Yes (weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' model) Python,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' MATLAB gumpy CNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' LSTM Keras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Theano Yes (weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' model) Python DeepEEG MLP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='3-D CNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' LSTM Keras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' TensorFlow Yes (weights) Python ExBrainable CNN PyTorch Yes (weights) Python Traja LSTM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' VAE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' GAN PyTorch Yes (weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' model) Python VARDNN toolbox Vector Auto-Regressive DNN Deep Learning Toolbox (MATLAB) Yes (weights) MATLAB Table 2: Model engineering specifications for the libraries and frameworks processing sequence data 9 Name Interface Online/Offline Maintenance Source BioPyC Jupyter Notebooks Offline Active gitlab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='fr/biopyc/BioPyC/ braindecode None Offline Active github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/braindecode/braindecode DeLINEATE GUI, Colab Notebooks Offline Active bitbucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='org/delineate/delineate EEG-DL None Offline Active github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/SuperBruceJia/EEG-DL gumpy None Online, Offline Inactive github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/gumpy-bci DeepEEG Colab Notebooks Offline Inactive github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/kylemath/DeepEEG ExBrainable GUI Offline Active github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/CECNL/ExBrainable Traja None Offline Active github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/traja-team/traja VARDNN toolbox None Offline Active github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/takuto-okuno-riken/vardnn Table 3: Technological aspects and code sources for the libraries and frameworks processing sequence data 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='2 Libraries for image data Libraries and frameworks for image data are shown in Tables 4 (domains of application), 5 (models characteristics),6 (technologies and sources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Computer vision and 2D image processing are arguably the fields in which DL has achieved the most impressive and state-of-art defining results, often inspiring and translating breakthroughs in other domanis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Classification and segmentation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' the separation of parts of the image based on their classes) are the most common tasks addressed by the image processing libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Magnetic resonance is the primary source of data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' however, various deep learning libraries are built microscopic and eye-tracking data as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Most of the libraries collected in our analysis take advantage of classical CNN architectures for classification, Convolutional AutoEncoders (CAEs) for segmentation, and GANs for synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' It is common to employ transfer learning to lessen the compu- tational and memory burden during the training phase, and take advantage of pre-trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Transfer learning consists in initializing models with parameters learnt on usually larger data sets, possibly from different domains and tasks, with varying amounts of further training in the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The best such examples are pose-estimation libraries extending the DeepLabCut system, arguably the most relevant project on the topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' DeepLabCut is an interactive framework for labelling, training, testing and refining models, that originally exploits the weights learned from ResNets (or newer architectures) on the ImageNet data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The results match human annotation using quite few training samples, holding for many (human and non-human) animals, and settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The documentation and demon- strative notebooks and tools offered by the Mathis Lab allow different levels of understanding and customization of the process, with high levels of robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Among the considered libraries, two set apart from the majority given the type of tasks they perform: GaNDLF addresses eXplainable AI (XAI), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Artificial Intelligence whose deci- sions and outputs can be understood by humans through more transparent mental models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' ANTsX performs both the co-registration step and super-resolution as a quality enhancing step for neuroimages, with the former being usually performed by traditional algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' GaNDLF sets its goal as the provision of deep learning resources in different layers of abstraction, allowing medical researchers with virtually no ML knowledge to perform robust experiments with models trained on carefully split data, with augmentations and preprocessing, under standardized protocols that can easily integrate interpretability tools such as Grad-CAM [53] and attention maps, which highlight the parts of an image according to how they influenced a model outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The ANTsX ecosystem is of similar wide scope, and is intended to build workflows on quantitative biology and medical imaging data, both in Python and R languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Packages from the same ecosystem perform registration of brain structures (by classical methods) as well as brain extraction by deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 11 Name Neuroscience area Data type Datasets Task AxonDeepSeg [54] Microbiology, Histology Img (SEM, TEM) External Segm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' DeepCINAC [55] Electrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Vid (2-photon calcium) No Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' DeepLabCut [56] Behavioral neuroscience Vid No Pose est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' DeepNeuro [57] Neuroimaging Img (fMRI, misc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=') No Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Segm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Synthesis DeepVOG [58] Oculography Img, Vid Demo Segm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' DeLINEATE [47] General Img, sequences External Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' DNNBrain [59] Brain map- ping Img No Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' ivadomed [60] Neuroimaging Img (2D, 3D) No Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Segm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' MEYE [61] Oculography Img, Vid Yes Segm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Allen Cell Structure Segmenter [62] Microbiology, Histology Img (3D-fluor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' microscopy) No Segm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' VesicleSeg [63] Microbiology, Histology Img (EM) No Segm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' CDeep3M2 [64] Microbiology, Histology Img (misc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' microscopy) Yes Segm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' CASCADE [65] Electrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Vid (2-photon calcium), Seq Yes Event detection ScLimibic [66] Neuroimaging Img (MRI) External Segm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' ALMA [67] Behavioral neuroscience Vid External Pose est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' fetal-code [68] Neuroimaging Img (rs-fMRI) External Segm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' ClinicaDL [69] Neuroimaging Img (MRI, PET) External Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Segm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' DeepNeuron [70] Microbiology, Histology Img (confocal microscopy) No Obj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Segm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' GaNDLF [71] Medical Imaging Img (2D, 3D) External Segm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Regression, XAI MesoNet [72] Neuroimaging Img (fluoresc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' microscopy) External Segm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Registration MARS, BENTO [73] Behavioral neuroscience Vid Yes Pose est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Action rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Tag NiftyNet [74] Medical Imaging Img (MRI, CT) No Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Segm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Synth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' ANTsX [75] (ANTsPyNet, ANTsRNet) Neuroimaging Img (MRI) No Classificastion, Segm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Registr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Super-res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' MARS, BENTO [73] Behavioral neuroscience Vid Yes Pose est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Action rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Tag Visual Fields Analysis [76] Eye tracking, Behavioral neuroscience Vid No Pose est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=', Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Table 4: Domains of applications for the libraries and frameworks processing image data 12 Name Models DL framework Customization Programming language AxonDeepSeg CAE TensorFlow Yes (weights) Python DeepCINAC DeepCINAC (CNN+LSTM) Keras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' TensorFlow Yes (weights) Python DeepLabCut CNN TensorFlow Yes (weights) Python DeepNeuro CNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' CAE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' GAN Keras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' TensorFlow Yes (weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' model) Python DeepVOG CAE TensorFlow No Python DeLINEATE CNN Keras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' TensorFlow Yes (weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' model) Python DNNBrain CNN PyTorch Yes (model) Python ivadomed 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='3-D CNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' CAE PyTorch Yes (weights,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' model) Python MEYE CAE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' CNN TensorFlow Yes (model) Python Allen Cell Structure Segmenter CAE PyTorch No Python VesicleSeg CNN PyTorch No Python CDeep3M2 CAE TensorFlow Yes (weights) Python CASCADE 1-D CNN TensorFlow Yes (weights) Python ScLimibic 3-D CAE neurite,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' TensorFlow No Python ALMA CNN Unspecified No Python fetal-code 2-D CNN TensorFlow No Python ClinicaDL CNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' CAE PyTorch Yes Python DeepNeuron CNN Unspecified No C++ GaNDLF CNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' CAE PyTorch Yes Python MesoNet CNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' CAE Keras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' TensorFlow No Python NiftyNet CNN TensorFlow Yes Python ANTsX (ANTsPyNet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' ANTsRNet) CNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' CAE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' GAN Keras,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' TensorFlow Yes Python,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' C++ MARS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' BENTO CNN TensorFlow Yes (weights) Python Visual Fields Analysis DeepLabCut TensorFlow,' metadata={'source': 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+page_content='com/saigerutherford/fetal-code ClinicaDL GUI, Colab Notebooks Offline Active github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/aramis-lab/clinicadl DeepNeuron GUI Online, Offline Inactive github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/Vaa3D/Vaa3D_Data/releases/tag/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='0 GaNDLF GUI Offline Active github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/CBICA/GaNDLF MesoNet GUI, Colab Notebooks Offline Active osf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='io/svztu NiftyNet None Offline Inactive github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/NifTK/NiftyNet ANTsX (ANTsPyNet, ANTsRNet) None Offline Active github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/ANTsX MARS, BENTO GUI, MATLAB GUI, Jupyter Notebooks Offline Active github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/neuroethology Visual Fields Analysis GUI Offline Active github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/mathjoss/VisualFieldsAnalysis Table 6: Technological aspects and code sources for the libraries and frameworks processing image data 14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='3 Libraries targeting data types different from sequences or images and general applications Libraries and frameworks for sequence data are shown in Tables 7 (domains of application), 8 (models characteris- tics), 9 (technologies and sources).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' In this category fall libraries and projects with either varying input data type, or other than sequence and image data analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' other libraries target computational platforms, higher hierarchy frame- works, or supporting functions for deep learning like specific preprocessing and augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' NeuroCAAS is an ambitious project that both standardizes experimental schedules, analyses and offers computational resources on the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The platform lifts the burden of configuring and deploying data analysis tool, guaranteeing also replicability and readily available usage of pre-made pipelines, with high efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' MONAI is a project that brings deep learning tools to many health and biology problems, and is a commonly used framework for the 3D variations of UNet [77] lately dominating the yearly BraTS challenge [32] (see at http://braintumorsegmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='org/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The paradigm builds on PyTorch and aims at unifying healthcare AI practices throughout both academia and enterprise research, not only in the model development but also in the creation of shared annotated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Lastly, it focuses on deployment and work in real world clinical production, settling as a strong candidate for being the standard solution in the do- main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Predify and THINGvision are two libraries that bridge deep learning research and computational neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The former allows to include an implementation of a «predictive coding mechanism» (as hypothesized in [78]) into virtually any pre-built architectures, evaluating its impact on performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The latter offers a single environment for Representational Similarity Analysis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' the study of the encodings of biological and artificial neural networks that process visual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 15 Name Neuroscience area Data type Datasets Task NeuroCAAS [79] Virtually all Virtually all External availability Virtually all MONAI [80] Virtually all Virtually all External availability Virtually all Predify [81] Computational Neuro- science Images,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Virtually all No Classification,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Adversarial attacks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' virtually all THINGvision [82] Computational Neuro- science Images,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Text External availability Classification TorchIO [83] Imaging All images No Augmentation Table 7: Domains of applications for the libraries and frameworks for special applications 16 Name Models DL framework Customization Programming language NeuroCAAS CNN TensorFlow Yes Python MONAI Virtually All PyTorch Yes Python Predify CNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Virtually all PyTorch Yes Python THINGvision CNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' RNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Transformers PyTorch,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' TensorFlow No Python TorchIO CNN PyTorch Yes Python Table 8: Model engineering specifications for the libraries and frameworks for special applications 17 Name Interface Online/Offline Maintenance Source NeuroCAAS GUI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Jupyter Notebooks Offline Active github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/cunningham-lab/neurocaas MONAI GUI, Colab Notebooks Offline Active github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/Project-MONAI/MONAI Predify Text UI (TOML) Offline Active github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/miladmozafari/predify THINGvision None Offline Active github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='com/ViCCo-Group/THINGSvision TorchIO GUI, Command line Offline Active torchio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='rtfd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='io Table 9: Technological aspects and code sources for the libraries and frameworks for special applications 18 5 Discussion The panorama of open-source libraries dedicated to deep learning applications in neuroscience is quite rich and diver- sified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' There is a corpus of organized packages that integrate preprocessing, training, testing and performance analyses of deep neural networks for neurological research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Most of these projects are tuned to specific data modalities and formats, but some libraries are quite versatile and customizabile, and there are projects that encompass quantitative biology and medical analysis as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' There is a common tendency to develop GUIs, enhancing user-friendliness of toolkits for non-programmers and researchers unacquainted with the command line interfaces, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Moreover, for the many libraries developed in Python, the (Jupyter) Notebook format appears as a widespread tool both for tutori- als, documentation and as an interface to cloud computational resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Google Colab [84]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Apart from specific papers and documentation, and outside of deep learning per se, it is important to make researchers and developers aware of the main topics and initiatives in open culture and Neuroinformatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' For this reason, the interested reader is invited to rely on competent institutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' INCF) and databases of open resources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' open-neuroscience) dedicated to Neuroscience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Among the possibly missing technologies, the queries employed did not retrieve results in Natural Language Processing libraries dedicated to neuroscience, nor toolkits specifically employing Graph Neural Networks (GNNs), although available in EEG-DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' NLP is actually fundamental in healthcare, since medical reports often come in non standardized forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Large language models, Named Entity Recognition (NER) systems and text mining approaches in biomedical research exist [85], [86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' GNNs comprise recent architectures that are extremely promising in a variety of fields [87], including biomedical research and particularly neuroscience [88], [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Even if promising, their application is still less mature than that of computer vision models or time series analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Considering the available software for imaging and signal processing in the domain of neuroscience, at this moment a single alternative targeting the opportunities offered by modern deep learning seems to be missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Overall, it seems still unlikely to develop a common deep learning framework for Neuroscience as a separate whole, but the engineering knowledge relevant and compressible into such framework would be common to other biomedical fields, and projects such as MONAI are strong candidates toward this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Instead, it seems achievable to deliver models and functions in a concerted way, restricted either to a sub-field or a data modality, based on the modularity of existent tools and the organizing possibilities of project initiation and management of open culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 6 Conclusions Although a large and growing number of repositories offer code to build specific models, as published in experimental papers, these resources seldom aim to constitute proper libraries or frameworks for research or clinical practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Both deep learning and neuroscience gain much value even from sophisticated proofs of concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' In parallel, organized packages are spreading and starting to provide and integrate pre-processing, training, testing and performance analyses of deep neural networks for neurological and biomedical research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' This paper has offered both an historical and a technical context for the use of deep neural networks in Neuroinformatics, focusing on open-source tools that scientists can comprehend and adapt to their necessities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' At the same time, this work underlines the value of the open culture and points to relevant institutions and platforms for neuroscientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Although the aim is not restricted to making clinicians develop their own deep models without coding or Machine Learning background, as was the case in [90], the overall effect of these libraries and sources is to democratize deep learning applications and results, as well as standardizing such complex and varied models, supporting the research community in obtaining proper means to an end, and in envisioning then realizing collectively new projects and tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Acknowledgments This work was supported by the "Department of excellence 2018-2022" initiative of the Italian Ministry of education (MIUR) awarded to the Department of Neuroscience - University of Padua.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' and Xiaoqiang Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' TensorFlow: Large-scale machine learning on heterogeneous systems, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Software available from tensorflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' [17] Francois Chollet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Keras, 2015.' metadata={'source': 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Walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Measurements of acute cerebral infarction: A clinical examination scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Stroke, 20(7):864–870, July 1989.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' [21] What is neuroinformatics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content='incf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} 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Korot, Joseph R Ledsam, Trevor Back, Reena Chopra, Nikolas Pontikos, Christoph Kern, Gabriella Moraes, Martin K Schmid, Dawn Sim, Konstantinos Balaskas, Lucas M Bachmann, Alastair K Denniston, and Pearse A Keane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' Automated deep learning design for medical image classification by health-care professionals with no coding experience: a feasibility study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' The Lancet Digital Health, 1(5):e232–e242, September 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} +page_content=' 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dE4T4oBgHgl3EQfZgyj/content/2301.05057v1.pdf'} diff --git a/0tFAT4oBgHgl3EQfCRy0/vector_store/index.pkl b/0tFAT4oBgHgl3EQfCRy0/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..b38058ab226b854bb882e0227ab80a9748ff666c --- /dev/null +++ b/0tFAT4oBgHgl3EQfCRy0/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d1d95d961160ea5f633a0068fcb664afd31353a3d5acd5e0ac1e88a3db2d4430 +size 347483 diff --git a/1tAyT4oBgHgl3EQf1flH/content/tmp_files/2301.00735v1.pdf.txt b/1tAyT4oBgHgl3EQf1flH/content/tmp_files/2301.00735v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b7636adbd7d34bacfc68348f1efebac149c0e804 --- /dev/null +++ b/1tAyT4oBgHgl3EQf1flH/content/tmp_files/2301.00735v1.pdf.txt @@ -0,0 +1,1622 @@ +arXiv:2301.00735v1 [math.DG] 2 Jan 2023 +FAILURE OF CURVATURE-DIMENSION CONDITIONS ON +SUB-RIEMANNIAN MANIFOLDS VIA TANGENT ISOMETRIES +LUCA RIZZI AND GIORGIO STEFANI +Abstract. We prove that, on any sub-Riemannian manifold endowed with a positive +smooth measure, the Bakry–Émery inequality for the corresponding sub-Laplacian, +1 +2∆(∥∇u∥2) ≥ g(∇u, ∇∆u) + K∥∇u∥2, +K ∈ R, +implies the existence of enough Killing vector fields on the tangent cone to force the latter +to be Euclidean at each point, yielding the failure of the curvature-dimension condition +in full generality. Our approach does not apply to non-strictly-positive measures. In +fact, we prove that the weighted Grushin plane does not satisfy any curvature-dimension +condition, but, nevertheless, does admit an a.e. pointwise version of the Bakry–Émery +inequality. +As recently observed by Pan and Montgomery, one half of the weighted +Grushin plane satisfies the RCD(0, N) condition, yielding a counterexample to gluing +theorems in the RCD setting. +1. Introduction and statements +In the last twenty years, there has been an impressive effort in extending the concept of +‘Ricci curvature lower bound’ to non-Riemannian structures, and even to general metric +spaces equipped with a measure (metric-measure spaces, for short). We refer the reader +to the ICM notes [3] for a survey of this line of research. +There are two distinct points of view on the matter, traditionally known as the La- +grangian and Eulerian approaches, respectively. +The Lagrangian point of view is the one adopted by Lott–Villani and Sturm [36, 48, +49]. In this formulation, Ricci curvature lower bounds are encoded by convexity-type +inequalities for entropy functionals on the Wasserstein space. Such inequalities are called +curvature-dimension conditions, CD(K, N) for short, where K ∈ R represents the lower +bound on the curvature and N ∈ [1, ∞] stands for an upper bound on the dimension. +The Eulerian point of view, instead, employs the metric-measure structure to define an +energy form and, in turn, an associated diffusion operator. The notion of Ricci curvature +lower bound is therefore encoded in the so-called Bakry–Émery inequality, BE(K, N) for +short, for the diffusion operator, which can be expressed in terms of a suitable Gamma +calculus, see the monograph [10]. +Thanks to several key contributions [4, 6, 7, 24], the Lagrangian and the Eulerian ap- +proaches are now known to be essentially equivalent. In particular, CD(K, N) always +Date: January 3, 2023. +2020 Mathematics Subject Classification. Primary 53C17. Secondary 54E45, 28A75. +Key words and phrases. Sub-Riemannian manifold, CD(K, ∞) condition, Bakry–Émery inequality, +infinitesimally Hilbertian, Grushin plane, privileged coordinates. +1 + +2 +L. RIZZI AND G. STEFANI +implies BE(K, N) in infinitesimal Hilbertian metric-measure spaces, as introduced in [25], +while the converse implication requires further technical assumptions. +Such synthetic theory of curvature-dimension conditions, besides being consistent with +the classical notions of Ricci curvature and dimension on smooth Riemannian manifolds, +is stable under pointed-measure Gromov–Hausdorff convergence. Furthermore, it yields +a comprehensive approach for establishing all results typically associated with Ricci cur- +vature lower bounds, like Poincaré, Sobolev, log-Sobolev and Gaussian isoperimetric in- +equalities, as well as Brunn–Minkowski, Bishop–Gromov and Bonnet–Myers inequalities. +1.1. The sub-Riemannian framework. Although the aforementioned synthetic cur- +vature-dimension conditions embed a large variety of metric-measure spaces, a relevant +and widely-studied class of smooth structures is left out—the family of sub-Riemmanian +manifolds. A sub-Riemannian structure is a natural generalization of a Riemannian one, +in the sense that its distance is induced by a scalar product that is defined only on a +smooth sub-bundle of the tangent bundle, whose rank possibly varies along the manifold. +See the monographs [2,40,45] for a detailed presentation. +The first result in this direction was obtained by Driver–Melcher [23], who proved that +an integrated version of the BE(K, ∞), the so-called pointwise gradient estimate for the +heat flow, is false for the three-dimensional Heisenberg group. +In [31], Juillet proved the failure of the CD(K, ∞) property for all Heisenberg groups +(and even for the strictly related Grushin plane, see [32]). Later, Juillet [33] extended his +result to any sub-Riemannian manifold endowed with a possibly rank-varying distribution +of rank strictly smaller than the manifold’s dimension, and with any positive smooth +measure, by exploiting the notion of ample curves introduced in [1]. The idea of [31,33] +is to construct a counterexample to the Brunn–Minkowski inequality. +The ‘no-CD theorem’ of [31] was extended to all Carnot groups by Ambrosio and the +second-named author in [8, Prop. 3.6] with a completely different technique, namely, by +exploiting the optimal version of the reverse Poincaré inequality obtained in [16]. +In the case of sub-Riemannian manifolds endowed with an equiregular distribution and +a positive smooth measure, Huang–Sun [29] proved the failure of the CD(K, N) condition +for all values of K ∈ R and N ∈ (1, ∞) contradicting a bi-Lipschitz embedding result. +Very recently, in order to address the structures left out in [33], Magnabosco–Rossi [37] +recently extended the ‘no-CD theorem’ to almost-Riemannian manifolds M of dimension 2 +or strongly regular. The approach of [37] relies on the localization technique developed by +Cavalletti–Mondino [19] in metric-measure spaces. +To complete the picture, we mention that several replacements for the Lott–Sturm– +Villani curvature-dimension property have been proposed and studied in the sub-Rieman- +nian framework in recent years. Far from being complete, we refer the reader to [11–15,38] +for an account on the Lagrangian approach, to [17] concerning the Eulerian one, and finally +to [47] for a first link between entropic inequalities and contraction properties of the heat +flow in the special setting of metric-measure groups. +Main aim. At the present stage, a ‘no-CD theorem’ for sub-Riemannian structures in +full generality is missing, since the aforementioned approaches [8,23,29,31,33,37] either +require the ambient space to satisfy some structural assumptions, or leave out the infinite +dimensional case N = ∞. + +FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS +3 +The main aim of the present paper is to fill this gap by showing that (possibly rank- +varying) sub-Riemannian manifolds do not satisfy any curvature bound in the sense of +Lott–Sturm–Villani or Bakry–Émery when equipped with a positive smooth measure, i.e., +a Radon measure whose density in local charts with respect to the Lebesgue measure is +a strictly positive smooth function. +1.2. Failure of the Bakry–Émery inequality. The starting point of our strategy is +the weakest curvature-dimension condition, as we now define. +Definition 1.1 (Bakry–Émery inequality). We say that a sub-Riemannian manifold +(M, d) endowed with a positive smooth measure m satisfies the Bakry–Émery BE(K, ∞) +inequality, for K ∈ R, if +1 +2 ∆(∥∇u∥2) ≥ g(∇u, ∇∆u) + K∥∇u∥2 +for all u ∈ C∞(M), +(1.1) +where ∆ is the corresponding sub-Laplacian, and ∇ the sub-Riemannian gradient. +Our first main result is the following rigidity property for sub-Riemannian structures +supporting the Bakry–Émery inequality (1.1). +Theorem 1.2 (no-BE). Let (M, d) be a complete sub-Riemannian manifold endowed +with a positive smooth measure m. If (M, d, m) satisfies the BE(K, ∞) inequality for some +K ∈ R, then rank Dx = dim M at each x ∈ M, so that (M, d) is Riemannian. +The idea behind our proof of Theorem 1.2 is to show that the metric tangent cone +in the sense of Gromov [26] at each point of (M, d) is Euclidean. This line of thought is +somehow reminiscent of the deep structural result for RCD(K, N) spaces, with K ∈ R and +N ∈ (1, ∞), proved by Mondino–Naber [39]. However, differently from [39], Theorem 1.2 +provides information about the metric tangent cone at each point of the manifold. Showing +that the distribution D is Riemannian at almost every point in fact would not be enough, +as this would not rule out almost-Riemannian structures. +Starting from (1.1), we first blow-up the sub-Riemannian structure and pass to its +metric-measure tangent cone, showing that (1.1) is preserved with K = 0. Note that, in +this blow-up procedure, the positivity of the density of m is crucial, since otherwise the +resulting metric tangent cone would be endowed with the null measure. +The resulting blown-up sub-Riemannian space is isometric to a homogeneous space +of the form G/H, where G = exp g is the Carnot group associated to the underlying +(finite-dimensional and stratified) Lie algebra g of bracket-generating vector fields, and +H = exp h is its subgroup corresponding to the Lie subalgebra h of vector fields vanishing +at the origin, see [18]. Of course, the most difficult case is when H is non-trivial, that is, +the tangent cone is not a Carnot group. +At this point, the key idea is to show that the Bakry–Émery inequality BE(K, ∞) +implies the existence of special isometries on the tangent cone. +Definition 1.3 (Sub-Riemannian isometries). Let M be a sub-Riemannian manifold, +with distribution D and metric g. A diffeomorphism φ : M → M is an isometry if +(φ∗D)|x = Dφ(x) +for all x ∈ M, +(1.2) +and, furthermore, φ∗ is an orthogonal map with respect to g. We say that a smooth vector +field V is Killing if its flow φV +t is an isometry for all t ∈ R. + +4 +L. RIZZI AND G. STEFANI +For precise definitions of g and h in the next statement, we refer to Section 2.4. +Theorem 1.4 (Existence of Killing fields). Let (M, d) be a complete sub-Riemannian +manifold equipped with a positive smooth measure m If (M, d, m) satisfies the BE(K, ∞) +inequality for some K ∈ R, then, for the nilpotent approximation at any given point, there +exists a vector space i ⊂ g1 such that +g1 = i ⊕ h1 +(1.3) +and every Y ∈ i is a Killing vector field. +The existence of the space of isometries i forces the Lie algebra g to be commutative and +of maximal rank, thus implying that the original manifold (M, d) was in fact Riemannian. +Theorem 1.5 (Killing implies commutativity). If there exists a subspace i ⊂ g1 of Killing +vector fields such that g1 = i ⊕ h1, then g is commutative. +Theorem 1.5 states that, if a Carnot group contains enough horizontal symmetries, then +it must be commutative. As it will be evident from its proof, Theorem 1.5 holds simply +assuming that, for each V ∈ i, the flow φV +t is pointwise distribution-preserving, namely it +satisfies (1.2), without being necessarily isometries. +1.3. Infinitesimal Hilbertianity. The Bakry–Émery inequality BE(K, ∞) in (1.1) is a +consequence of the CD(K, ∞) condition as soon as the ambient metric-measure space is +infinitesimal Hilbertian as defined in [25]. +Let (X, d) be a complete separable metric space, m be a locally bounded Borel mea- +sure, and q ∈ [1, ∞). We let |Du|w,q ∈ Lq(X, m) be the minimal q-upper gradient of a +measurable function u : X → R, see [5, Sec. 4.4]. We define the Banach space +W1,q(X, d, m) = {u ∈ Lq(X, m) : |Du|w,q ∈ Lq(X, m)} +with the norm +∥u∥W1,q(X,d,m) = +� +∥u∥q +Lq(X,m) + ∥|Du|w,q∥q +Lq(X,m) +�1/q . +Definition 1.6 (Infinitesimal Hilbertianity). A metric measure space (X, d, m) is in- +finitesimally Hilbertian if W1,2(X, d, m) is a Hilbert space. +The infinitesimal Hilbertianity of sub-Riemannian structures has been recently proved +in [35], with respect to any Radon measure. +In particular, Theorem 1.2 immediately +yields the following ‘no-CD theorem’ for sub-Riemannian manifolds, thus extending all +the aforementioned results [8,23,29,31,33,37]. +Corollary 1.7 (no-CD). Let (M, d) be a complete sub-Riemannian manifold endowed +with a positive smooth measure m. If (M, d, m) satisfies the CD(K, ∞) condition for some +K ∈ R, then (M, d) is Riemannian. +However, since the measure in Corollary 1.7 is positive and smooth, we can avoid to +rely on the general result of [35], instead providing a simpler and self-contained proof +of the infinitesimal Hilbertianity property. In particular, we prove the following result, +which actually refines [35, Th. 5.6] in the case of smooth measures. In the following, +HW1,q(M, m) denotes the sub-Riemannian Sobolev spaces (see Section 2.2). + +FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS +5 +Theorem 1.8 (Infinitesimal Hilbertianity). Let q ∈ (1, ∞). Let (M, d) be a complete sub- +Riemannian manifold equipped with a positive smooth measure m. The following hold. +(i) W1,q(M, d, m) = HW1,q(M, m), with |Du|w,q = ∥∇u∥ m-a.e. on M for all u ∈ +W1,q(M, d, m). In particular, taking q = 2, (M, d, m) is infinitesimally Hilbertian. +(ii) If (M, d, m) satisfies the CD(K, ∞) condition for some K ∈ R, then the Bakry– +Émery BE(K, ∞) inequality (1.1) holds on M. +Note that Theorem 1.8 holds for less regular measures, see Remark 3.6. +Remark 1.9 (The case of a.e. smooth measures). Theorem 1.8 can be adapted also to +the case of a Borel and locally finite measure m which is smooth and positive only on Ω, +where Ω ⊂ M is an open set with m(∂Ω) = 0. In this case, we obtain HW1,q(Ω, m) = +W1,q(Ω, d, m), with |Du|w,q = ∥∇u∥ m-a.e. on Ω for all u ∈ W1,q(Ω, d, m). In particular, +if m is smooth and positive out of a closed set Z, with m(Z) = 0, an elementary ap- +proximation argument proves that (M, d, m) is infinitesimally Hilbertian and, if (M, d, m) +satisfies the CD(K, ∞) condition for K ∈ R, then the Bakry-Émery BE(K, ∞) inequality +(1.1) holds on M \Z. This is the case, for example, of the Grushin planes and half-planes +with weighted measures of Section 1.5. The proof follows the same argument of the one of +Theorem 1.8, exploiting the locality of the q-upper gradient, see for example [5, Sec. 8.2] +and [25, Prop. 2.6], and similar properties for the distributional derivative. +1.4. An alternative approach to the ‘no-CD theorem’. We mention an alternative +proof of the ‘no-CD theorem’ for almost-Riemannian structures (i.e., sub-Riemannian +structures that are Riemannian outside a closed nowhere dense singular set). The strategy +relies on the Gromov-Hausdorff continuity of the metric tangent at interior points of +geodesics in RCD(K, N) spaces, with N < ∞, proved by Deng in [22], +For example, consider the standard Grushin plane (introduced in Section 1.5) equipped +with a smooth positive measure. The curve γ(t) = (t, 0), t ∈ R, is a geodesic between +any two of its point. The metric tangent at γ(t) is (isometric to) the Euclidean plane for +every t ̸= 0, while it is (isometric to) the Grushin plane itself for t = 0. Since the Grushin +plane cannot be bi-Lipschitz embedded into the Euclidean plane, the two spaces are at +positive Gromov-Hausdorff distance, contradicting the continuity result. +This strategy has a few drawbacks. +On the one hand, it relies on the (non-trivial) +machinery developed in [22]. +Consequently, this argument does not work in the case +N = ∞. On the other hand, the formalization of this strategy for general almost-Rie- +mannian structures requires certain quantitative bi-Lipschitz non-embedding results for +almost-Riemannian structures into Euclidean spaces, which we are able to prove only +under the same assumptions of [37]. +1.5. Weighted Grushin structures. When the density of the smooth measure is al- +lowed to vanish, the ‘no-CD theorem’ breaks down. In fact, in this situation, the following +two interesting phenomena occur: +(A) the Bakry-Émery BE(K, ∞) inequality no longer implies the CD(K, ∞) condition; +(B) there exist almost-Riemannian structures with boundary satisfying the CD(0, N) +condition for N ∈ [1, ∞]. + +6 +L. RIZZI AND G. STEFANI +We provide examples of both phenomena on the so-called weighted Grushin plane. This +is the sub-Riemannian structure on R2 induced by the family F = {X, Y }, where +X = ∂x, +Y = x ∂y, +(x, y) ∈ R2. +(1.4) +The induced distribution D = span{X, Y } has maximal rank outside the singular region +S = {x = 0} and rank 1 on S. Since [X, Y ] = ∂y on R2, the resulting sub-Riemannian +metric space (R2, d) is Polish and geodesic. It is almost-Riemannian in the sense that, out +of S, the metric is locally equivalent to the Riemannian one given by the metric tensor +g = dx ⊗ dx + 1 +x2 dy ⊗ dy, +x ̸= 0. +(1.5) +We endow the metric space (R2, d) with the weighted Lebesgue measure +mp = |x|p dx dy, +where p ∈ R is a parameter. The choice p = −1 corresponds to the Riemannian density +volg = 1 +|x| dx dy, +x ̸= 0, +(1.6) +so that +mp = e−V volg, +V (x) = −(p + 1) log |x|, +x ̸= 0. +(1.7) +We call the metric-measure space Gp = (R2, d, mp) the (p-)weighted Grushin plane. +We can now state the following result, illustrating phenomenon (A). +Theorem 1.10. Let p ∈ R and let Gp = (R2, d, mp) be the weighted Grushin plane. +(i) If p ≥ 0, then Gp does not satisfy the CD(K, ∞) property for all K ∈ R. +(ii) If p ≥ 1, then Gp satisfies the BE(0, ∞) inequality (1.1) almost everywhere. +To prove (i), we show that the corresponding Brunn–Minkowski inequality is violated. +In fact, the case p = 0 is due to Juillet [32], while the case p > 0 can be achieved via a +simple argument which was pointed out to us by J. Pan. Claim (ii), instead, is obtained +by direct computations. +Somewhat surprisingly, the weighted Grushin half -plane G+ +p —obtained by restricting +the metric-measure structure of Gp to the (closed) half-plane [0, ∞)×R—does satisfy the +CD(0, N) condition for sufficiently large N ∈ [1, ∞]. Precisely, we can prove the following +result, illustrating phenomenon (B). +Theorem 1.11. Let p ≥ 1. The weighted Grushin half-plane G+ +p satisfies the CD(0, N) +condition if and only if N ≥ Np, where Np ∈ (2, ∞] is given by +Np = (p + 1)2 +p − 1 ++ 2, +(1.8) +with the convention that N1 = ∞. Furthermore, G+ +p is infinitesimally Hilbertian, and it +is thus an RCD(0, N) space for N ≥ Np. +While we were completing this work, Pan and Montgomery [41] observed that the spaces +built in [20, 42] as Ricci limits are actually the weighted Grushin half-spaces presented +above. Our construction and method of proof are more direct with respect to the approach +of [20,42], and easily yield sharp dimensional bounds. + +FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS +7 +1.6. Counterexample to gluing theorems. We end this introduction with an inter- +esting by-product of our analysis, in in connection with the so-called gluing theorems. +Perelman’s Doubling Theorem [43, Sect. 5.2] states that a finite dimensional Alexan- +drov space with a curvature lower bound can be doubled along its boundary yielding an +Alexandrov space with same curvature lower bound and dimension. This result has been +extended by Petrunin [44, Th. 2.1] to the gluing of Alexandrov spaces. +It is interesting to understand whether these classical results hold true for general +metric-measure spaces satisfying synthetic Ricci curvature lower bounds in the sense of +Lott–Sturm–Villani. In [34], the gluing theorem was proved for CD(K, N) spaces with +Alexandrov curvature bounded from below (while it is false for MCP spaces, see [46]). +Here we obtain that, in general, the assumption of Alexandrov curvature bounded +from below cannot be removed from the results in [34]. More precisely, Theorems 1.10 +and 1.11, and the fact that the metric-measure double of the Grushin half-plane G+ +p is Gp +(see [46, Prop. 6]) yield the following corollary. +Corollary 1.12 (Counterexample to gluing in RCD spaces). For all N ≥ 10, there exists +a geodesically convex RCD(0, N) metric-measure space with boundary such that its metric- +measure double does not satisfy the CD(K, ∞) condition for any K ∈ R. +In [34, Conj. 1.6], the authors conjecture the validity of the gluing theorem for non- +collapsed RCD(K, N), with N the Hausdorff dimension of the metric-measure space. +As introduced in [21], a non-collapsed RCD(K, N) space is an infinitesimally Hilbertian +CD(K, N) space with m = H N, where H N denotes the N-dimensional Hausdorff mea- +sure of (X, d). Since the weighted half-Grushin spaces are indeed collapsed, Corollary 1.12 +also shows that the non-collapsing assumption cannot be removed from [34, Conj. 1.6]. +1.7. Acknowledgments. We wish to thank Michel Bonnefont for fruitful discussions +and, in particular, for bringing some technical details in [23] that inspired the strategy of +the proof of Theorem 1.2 to our attention. +This work has received funding from the European Research Council (ERC) under the +European Union’s Horizon 2020 research and innovation programme (grant agreement No. +945655) and the ANR grant ‘RAGE’ (ANR-18-CE40-0012). The second-named author +is member of the Istituto Nazionale di Alta Matematica (INdAM), Gruppo Nazionale +per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA), and is par- +tially supported by the INdAM–GNAMPA 2022 Project Analisi geometrica in strutture +subriemanniane, codice CUP_E55F22000270001. +2. Preliminaries +In this section, we introduce some notation and recall some results about sub-Rieman- +nian manifolds and curvature-dimension conditions. +2.1. Sub-Riemannian structures. For L ∈ N, we let F = {X1, . . . , XL} be a family +of smooth vector fields globally defined on a smooth n-dimensional manifold M, n ≥ 2. +The (generalized) sub-Riemannian distribution induced by the family F is defined by +D = +� +x∈M +Dx, +Dx = span{X1|x, . . . , XL|x} ⊂ TxM, +x ∈ M. +(2.1) + +8 +L. RIZZI AND G. STEFANI +Note that we do not require the dimension of Dx to be constant as x ∈ M varies, that is, +we may consider rank-varying distributions. With a standard abuse of notation, we let +Γ(D) = C∞-module generated by F. +Notice that, for any smooth vector field V , it holds +V ∈ Γ(D) =⇒ Vx ∈ Dx for all x ∈ M, +but the converse is false in general. We let +∥V ∥x = min +� +|u| : u ∈ RL such that V = +L +� +i=1 +ui Xi|x, Xi ∈ F +� +(2.2) +whenever V ∈ D and x ∈ M. The norm ∥ · ∥x induced by the family F satisfies the +parallelogram law and, consequently, it is induced by a scalar product +gx : Dx × Dx → R. +An admissible curve is a locally Lipschitz in charts path γ : [0, 1] → M such that there +exists a control u ∈ L∞([0, 1]; RL) such that +˙γ(t) = +L +� +i=1 +ui(t)Xi|γ(t) +for a.e. t ∈ [0, 1]. +The length of an admissible curve γ is defined via the norm (2.2) as +length(γ) = +� 1 +0 ∥˙γ(t)∥γ(t) dt +and the Carnot–Carathéodory (or sub-Riemannian) distance between x, y ∈ M is +d(x, y) = inf{length(γ) : γ admissible with γ(0) = x, γ(1) = y}. +We assume that the family F satisfies the bracket-generating condition +TxM = {X|x : X ∈ Lie(F)} +for all x ∈ M, +(2.3) +where Lie(F) is the smallest Lie subalgebra of vector fields on M containing F, namely, +Lie(F) = span +� +[Xi1, . . . , [Xij−1, Xij]] : Xiℓ ∈ F, j ∈ N +� +. +Under the assumption (2.3), the Chow–Rashevskii Theorem implies that d is a well-defined +finite distance on M inducing the same topology of the ambient manifold. +2.2. Gradient, sub-Laplacian and Sobolev spaces. The gradient of a function u ∈ +C∞(M) is the unique vector field ∇u ∈ Γ(D) such that +g(∇u, V ) = du(V ) +for all V ∈ Γ(D). +(2.4) +One can check that ∇u can be globally represented as +∇u = +L +� +i=1 +Xiu Xi, +with +∥∇u∥2 = +L +� +i=1 +(Xiu)2, +(2.5) +even if the family F is not linearly independent, see Corollary A.2 for a proof. + +FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS +9 +We equip the manifold M with a positive smooth measure m. The sub-Laplacian of a +function u ∈ C∞(M) is the unique function ∆u ∈ C∞(M) such that +� +M g(∇u, ∇v) dm = − +� +M v ∆u dm +(2.6) +for all v ∈ C∞ +c (M). On can check that ∆u can be globally represented as +∆u = +L +� +i=1 +� +X2 +i u + Xiu divm(Xi) +� +, +(2.7) +see Corollary A.2 for a proof. In (2.7), divmV is the divergence of the vector field V +computed with respect to m, that is, +� +M v divm(V ) dm = − +� +M g(∇v, V ) dm +for all v ∈ C∞ +c (M). +For q ∈ [1, ∞), we say that u ∈ L1 +loc(M, m) has q-integrable distributional Xi-derivative +if there exists a function Xiu ∈ Lq(M, m) such that +� +M vXiu dm = +� +M uX∗ +i v dm +for all v ∈ C∞ +c (M), +where X∗ +i v = −Xiv − v divm(Xi) denotes the adjoint action of Xi. We thus let +HW1,q(M, m) = {u ∈ Lq(M, m) : Xiu ∈ Lq(M, m), i = 1, . . ., L} +be the usual horizontal W1,q Sobolev space induced by the the family F and the measure m +on M, endowed with the natural norm +∥u∥HW1,q(M,m) = +� +∥u∥q +Lq(M,m) + ∥∇u∥q +Lq(M,m) +�1/q +for all u ∈ HW1,q(M, m), where ∇u = +L +� +i=1 +Xiu Xi in accordance with (2.5) and +∥∇u∥q +Lq(M,m) = +� +M ∥∇u∥q dm. +2.3. Privileged coordinates. Following [18,30], we introduce privileged coordinates, a +fundamental tool in the description of the tangent cone of sub-Riemannian manifolds. +Given a multi-index I ∈ {1, . . ., L}×i, i ∈ N, we let |I| = i be its length and we set +XI = [XI1, [. . ., [XIi−1, XIi]]]]. +Accordingly, we define +Di +x = span{XI|x : |I| ≤ i} +(2.8) +and +ki(x) = dim Di +x +for all x ∈ M and i ∈ N. In particular, D0 +x = {0} and D1 +x = Dx as in (2.1) for all x ∈ M. +The spaces defined in (2.8) naturally yield the filtration +{0} = D0 +x ⊂ D1 +x ⊂ · · · ⊂ Ds(x) +x += TxM +for all x ∈ M, where s = s(x) ∈ N is the step of the sub-Riemannian structure at the +point x. We say that x ∈ M is a regular point if the dimension of each space Di +y remains +constant as y ∈ M varies in an open neighborhood of x, otherwise x is a singular point. + +10 +L. RIZZI AND G. STEFANI +Definition 2.1 (Adapted and privileged coordinates). Let o ∈ M and let U ⊂ M be an +open neighborhood of o. We say that the local coordinates given by a diffeomorphism +z : U → Rn are adapted at o if they are centered at o, i.e. z(o) = 0, and ∂z1|0, . . ., ∂zki|0 +form a basis for Di +o in these coordinates for all i = 1, . . ., s(o). We say that the adapted +coordinate zi has weight wi = j if ∂zi|0 ∈ Dj +o \ Dj−1 +o +. Furthermore, we say that the coor- +dinates z are privileged at o if they are adapted at o and, in addition, zi(x) = O(d(x, o)wi) +for all x ∈ U and i = 1, . . ., n. +Privileged coordinates exist in a neighborhood of any point, see [18, Th. 4.15]. +2.4. Nilpotent approximation. From now on, we fix a set of privileged coordinates +z : U → Rn around a point o ∈ M in the sense of Definition 2.1. +Without loss of +generality, we identify the coordinate domain U ⊂ M with Rn and the base point o ∈ M +with the origin 0 ∈ Rn. Similarly, the vector fields in F defined on U are identified with +vector fields on Rn, and the restriction of the sub-Riemannian distance d to U is identified +with a distance function on Rn, which is induced by the family F, for which we keep the +same notation. +On (Rn, F), we define a family of dilations, for λ ≥ 0, by letting +dilλ : Rn → Rn, +dilλ(z1, . . . , zn) = (λw1z1, . . . , λwnzn) +for all z = (z1, . . . , zn) ∈ Rn, where the wi’s are the weights given by Definition 2.1. We +say that a differential operator P is homogeneous of degree −d ∈ Z if +P(f ◦ dilλ) = λ−d(Pf) ◦ dilλ +for all λ > 0 and f ∈ C∞(Rn). +(2.9) +Note that the monomial zi is homogeneous of degree wi, while the vector field ∂zi is +homogeneous of degree −wi, for i = 1, . . . , n. As a consequence, the differential operator +zµ1 +1 · · · · · zµn +n +∂|ν| +∂zν1 +1 · · · ∂zνn +n +, +νi, µj ∈ N ∪ {0}, +is homogeneous of degree �n +i=1 wi(µi − νi). For more details, see [18, Sec. 5]. +We can now introduce the new family +� +F = +�� +X1, . . . , � +XL +� +by defining +� +Xi = lim +ε→0 Xε +i , +Xε +i = ε (dil1/ε)∗Xi, +(2.10) +for all i = 1, . . . , L, where (dil1/ε)∗ stands for the usual push-forward via the differential +of the dilation map dil1/ε, see [18, Sec. 5.3]. The convergence in (2.10) can be actually +made more precise, in the sense that +Xε +i = � +Xi + Rε +i, +i = 1, . . ., L, +where Rε +i locally uniformly converges to zero as ε → 0, see [18, Th. 5.19]. +The family � +F is a set of complete vector fields on Rn, homogeneous of degree −1, with +polynomial coefficients, and can be understood as the ‘principal part’ of F upon blow-up +by dilations. Since F satisfies the bracket-generating condition (2.3), also the new family +� +F is bracket-generating at all points of Rn, and thus induces a finite sub-Riemannian +distance �d, see [18, Prop. 5.17]. The resulting n-dimensional sub-Riemannian structure +(Rn, � +F ) is called nilpotent approximation of (Rn, F) at 0 ∈ Rn. + +FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS +11 +The family � +F = +�� +X1, . . ., � +XL +� +generates a finite-dimensional stratified Lie algebra +g = Lie( � +F ) = g1 ⊕ · · · ⊕ gs +of step s = s(0) ∈ N, where the grading is given by the degree of the vector fields, +according to the definition in (2.9), that is, the layer gi corresponds to vector fields +homogeneous of degree −i with respect to dilations, see [18, Sec. 5.4]. +In particular, +g1 = span +�� +X1, . . . , � +XL +� +, so that g is generated by its first stratum, namely, +gj+1 = [g1, gj], +∀j = 1, . . . , s − 1. +(2.11) +Finally, define the Lie subalgebra of vector fields vanishing at 0, +h = +�� +X ∈ g : � +X|0 = 0 +� += h1 ⊕ · · · ⊕ hs, +which inherits the grading from the one of g, +hj+1 = [h1, hj], +∀j = 1, . . ., s − 1. +(2.12) +It is a fundamental fact [18, Th. 5.21] that the nilpotent approximation (Rn, � +F ) is diffeo- +morphic to the homogeneous sub-Riemannian space G/H, where G is the Carnot group +G = exp g (explicitly realized as the subgroup of the flows of the vector fields of g acting +on Rn from the right) and H = exp h is the Carnot subgroup induced by h. +In particular, if 0 ∈ Rn is a regular point, then H = {0}, and so the nilpotent approxi- +mation (Rn, � +F ) is diffeomorphic to the Carnot group G, see [18, Prop. 5.22]. +Recall that the smooth measure m on the original manifold M can be identified with +a smooth measure on U ≃ Rn, for which we keep the same notation. In particular, m +is absolutely continuous with respect to the n-dimensional Lebesgue measure L n on Rn, +with m = ρ L n for some positive smooth function ρ: Rn → (0, ∞). The corresponding +blow-up measure on the nilpotent approximation is naturally given by +�m = lim +ε→0 mε = ρ(0) L n, +mε = εQ (dil1/ε)#m, +in the sense of weak∗ convergence of measures in Rn, where +Q = +n +� +i=1 +i wi ∈ N +is the so-called homogeneous dimension of (Rn, � +F ) and (dil1/ε)# stands for the push- +forward in the measure-theoretic sense via the dilation map dil1/ε. Consequently, without +loss of generality, we can assume that ρ(0) = 1, thus endowing (Rn, � +F ) with the n- +dimensional Lebesgue measure. +Notice that divL n � +Xi = 0, for all i = 1, . . . , L, since +each � +Xi is homogeneous of degree −1. Hence, by (2.7), the sub-Laplacian of a function +u ∈ C∞(Rn) can be globally represented as +�∆u = +L +� +i=1 +� +X 2 +i u. +(2.13) +It is worth noticing that the metric space (Rn, �d ) induced by the nilpotent approxi- +mation (Rn, � +F ) actually coincides with the metric tangent cone at o ∈ M of the metric +space (M, d) in the sense of Gromov [26], see [18, Th. 7.36] for the precise statement. + +12 +L. RIZZI AND G. STEFANI +In fact, the sub-Riemmanian distance dε induced by the vector fields Xε +i , i = 1, . . . , L, +defined in (2.10) is uniformly converging to the distance �d on compact sets as ε → 0. +It is not difficult to check that the family {(Rn, dε, mε, 0)}ε>0 of pointed metric-measure +spaces converge to the pointed metric-measure space (Rn, �d, L n, 0) as ε → 0 in the pointed +measure Gromov–Hausdorff topology, see [13, Sec. 10.3] for details. +2.5. The curvature-dimension condition. We end this section by recalling the defi- +nition of curvature-dimension conditions of introduced in [36,48,49]. +On a Polish (i.e., separable and complete) metric space (X, d), we let P(X) be the set +of probability Borel measures on X and define the Wasserstein (extended) distance W2 +W2 +2(µ, ν) = inf +�� +X×X d2(x, y) dπ : π ∈ Plan(µ, ν) +� +∈ [0, ∞], +for µ, ν ∈ P(X), where +Plan(µ, ν) = {π ∈ P(X × X) : (p1)#π = µ, (p2)#π = ν}, +where pi : X ×X → X, i = 1, 2, are the projections on each component and T#µ ∈ P(Y ) +denotes the push-forward measure given by any µ-measurable map T : X → X. The +function W2 is a distance on the Wasserstein space +P2(X) = +� +µ ∈ P(X) : +� +X d2(x, x0) dµ(x) < ∞ for some, and thus any, x0 ∈ X +� +. +Note that (P2(X), W2) is a Polish metric space which is geodesic as soon as (X, d) is. In +addition, letting Geo(X) be the set of geodesics of (X, d), namely, curves γ : [0, 1] → X +such that d(γs, γt) = |s−t| d(γ0, γ1), for all s, t ∈ [0, 1], any W2-geodesic µ: [0, 1] → P2(X) +can be (possibly non-uniquely) represented as µt = (et)♯ν for some ν ∈ P(Geo(X)), where +et: Geo(X) → X is the evaluation map at time t ∈ [0, 1]. +We endow the metric space (X, d) with a non-negative Borel measure m such that +m is finite on bounded sets and supp(m) = X. +We define the (relative) entropy functional Entm : P2(X) → [−∞, +∞] by letting +Entm(µ) = +� +X ρ log ρ dm +if µ = ρm and ρ log ρ ∈ L1(X, m), while we set Entm(µ) = +∞ otherwise. +Definition 2.2 (CD(K, ∞) property). We say that a metric-measure space (X, d, m) +satisfies the CD(K, ∞) property if, for any µ0, µ1 ∈ P2(X) with Entm(µi) < +∞, i = 0, 1, +there exists a W2-geodesic [0, 1] ∋ s �→ µs ∈ P2(X) joining them such that +Entm(µs) ≤ (1 − s) Entm(µ0) + s Entm(µ1) − K +2 s(1 − s) W2 +2(µ0, µ1) +(2.14) +for every s ∈ [0, 1]. +The geodesic K-convexity of Entm in (2.14) can be reinforced to additionally encode an +upper bound on the dimension on the space, as recalled below. For N ∈ (1, ∞), we let +SN(µ, m) = − +� +X ρ−1/N dµ, +µ = ρm + µ⊥, +be the N-Rényi entropy of µ ∈ P2(X) with respect to m, where µ = ρm + µ⊥ denotes +the Radon–Nikodym decomposition of µ with respect to m. + +FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS +13 +Definition 2.3 (CD(K, N) property). We say that a metric-measure space (X, d, m) +satisfies the CD(K, N) property for some N ∈ [1, ∞) if, for any µ0, µ1 ∈ P2(X) with +µi = ρim, i = 0, 1, there exists a W2-geodesic [0, 1] ∋ s �→ µs ∈ P2(X) joining them, with +µs = (es)♯ν for some ν ∈ P(Geo(X)) such that +SN′(µs, m) ≤ − +� +Geo(X) +� +τ (1−s) +K,N′ (d(γ0, γ1))ρ−1/N′ +0 +(γ0) + τ (s) +K,N′(d(γ0, γ1))ρ−1/N′ +1 +(γ1) +� +dν(γ) +for every s ∈ [0, 1], N′ ≥ N. Here τ (s) +K,N is the model distortion coefficient, see [49, p. 137]. +Remark 2.4. The CD(0, N) corresponds to the convexity of the N′-Rényi entropy +SN′(µs, m) ≤ (1 − s)SN′(µ0, m) + sSN′(µ1, m), +for every s ∈ [0, 1] and N′ ≥ N, with µ0, µ1 ∈ P2(X) as in Definition 2.3. +Remark 2.5. For a CD(K, N) metric-measure space, K and N represent a lower bound +on the Ricci tensor and an upper bond on the dimension, respectively, and we have +CD(K, N) =⇒ CD(K, N′) +for all N′ ≥ N, N, N′ ∈ [1, ∞], +CD(K, N) =⇒ CD(K′, N) +for all K′ ≤ K, K, K′ ∈ R. +In particular, the CD(K, ∞) condition (2.14) is the weakest of all the curvature-dimension +conditions for fixed K ∈ R. +3. Proofs +We first deal with Theorems 1.4 and 1.5, from which Theorem 1.2 immediately follows. +3.1. Proof of Theorem 1.4. We divide the proof in four steps. +Step 1: passing to the nilpotent approximation via blow-up. Let (Rn, � +F ) be the nilpotent +approximation of (M, F) at some fixed point o ∈ M as explained in Section 2.4. Let +u ∈ C∞ +c (M) and, without loss of generality, let us assume that supp u is contained in the +domain of the privileged coordinates at o ∈ M. In particular, we identify u with a C∞ +c +function on Rn. We now apply (1.1) to the dilated function +uε = u ◦ dil1/ε ∈ C∞ +c (Rn), +for ε > 0, +and evaluate this expression at the point dilε(x) ∈ Rn. Exploiting the expressions in Corol- +lary A.2, we get that +L +� +i,j=1 +Xε +i u +� +Xε +ijju − Xε +jjiu +� +− (Xε +iju)2 + Rε +i,j u ≤ 0, +(3.1) +where Xε +i is as in (2.10) , Xijk = XiXjXk whenever i, j, k ∈ {1, . . . , L}, and Rε +i,j is a +reminder locally uniformly vanishing as ε → 0. Therefore, letting ε → 0 in (3.1), by the +convergence in (2.10) we get +L +� +i,j=1 +� +Xiu +�� +Xijju − � +Xjjiu +� +− +�� +Xiju +�2 ≤ 0, +(3.2) +which is (1.1) with K = 0 for the nilpotent approximation (Rn, � +F ). + +14 +L. RIZZI AND G. STEFANI +Step 2: improvement via homogeneous structure. We now show that (3.2) implies a +stronger identity, see (3.4) below, obtained from (3.2) by removing the squared term and +replacing the inequality with an equality. Recall, in particular, the definition of weight of +(privileged) coordinates in Definition 2.1. We take u ∈ C∞(Rn) of the form +u = α + γ, +where α and γ are homogeneous polynomial of weighted degree 1 and at least 3, respec- +tively. Since XIα = 0 as soon as the multi-index satisfies |I| ≥ 2 (see [18, Prop. 4.10]), +we can take the terms with lowest homogeneous degree in (3.2) to get +L +� +i,j=1 +� +Xiα +�� +Xijjγ − � +Xjjiγ +� += +L +� +i=1 +� +Xiα +�� +Xi, �∆ +� +(γ) ≤ 0 +for all such α and γ. In the second equality, we used the fact that the sub-Laplacian �∆ is +a sum of squares as in (2.13). Since α can be replaced with −α, we must have that +L +� +i=1 +� +Xiα +�� +Xi, �∆ +� +(γ) = 0. +(3.3) +Observing that � +Xiα is homogeneous of degree 0, and thus a constant function, we can +rewrite (3.3) as +� L +� +i=1 +� +Xiα � +Xi, �∆ +� +(γ) = 0, +(3.4) +which is the seeked improvement of (3.2). +Step 3: construction of the space i ⊂ g1. Let Pn +1 be the vector space of homogeneous +polynomials of weighted degree 1 on Rn. Notice that +Pn +1 = span{zi | i = 1, . . . , k1}, +k1 = dim D|0, +that is, Pn +1 is generated by the monomials given by the coordinates of lowest weight. We +now define a linear map φ: Pn +1 → g1 by letting +φ[α] = � +∇α = +L +� +i=1 +� +Xiα � +Xi +for all α ∈ Pn +1 (recall Corollary A.2). We claim that φ is injective. Indeed, if φ[α] = 0 for +some α ∈ Pn +1, then, by applying the operator φ[α] to the polynomial α, we get +0 = φ[α](α) = +� L +� +i=1 +� +Xiα � +Xi +� +(α) = +L +� +i=1 +(� +Xiα)2. +Thus � +Xiα = 0 for all i = 1, . . ., L. +Hence α must have weighted degree at least 2. +However, since α is homogeneous of weighted degree 1, we conclude that α = 0, proving +that ker φ = {0}. We can thus define the subspace +i = φ[Pn +1] ⊂ g1. + +FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS +15 +By (3.4), any � +X ∈ i is such that [� +X, �∆](γ) = 0 for any homogeneous polynomial γ +of degree at least 3. Exploiting the definitions given in Section 2.4, we observe that a +differential operator P, homogeneous of weighted degree −d ∈ Z, has the form +P = +� +µ,ν +aµ,νzµ ∂|ν| +∂zν , +(3.5) +where µ = (µ1, . . . , µn), ν = (ν1, . . . , νn), µi, νj ∈ N ∪ {0}, aµ,ν ∈ R, and the weighted +degree of every addend in (3.5) is equal to −d, namely, �n +i=1(µi − νi)wi = −d. +Thus, since � +X and �∆ are homogeneous differential operators of order −1 and −2, +respectively, then [� +X, �∆] has order −3, see [18, Prop. 5.16]. It follows that [� +X, �∆] = 0 as +differential operator acting on C∞(Rn). +We now show (1.3). Let us first observe that i ∩ h = {0}. Indeed, if φ[α] ∈ h for some +α ∈ Pn +1, that is, φ[α]|0 = 0, then � +Xiα|0 = 0 for all i = 1, . . ., L. Since � +Xiα is a constant +function, this implies φ[α] = 0, as claimed. Therefore, since dim i = dim Pn +1 = k1, we must +have g1 = i ⊕ h1 thanks to Lemma 3.1 below. +Lemma 3.1. With the same notation of Section 2.4, if g1 = v ⊕ h1, then dim v = k1. +Proof. We claim that the dimension of v is preserved by evaluation at zero, that is, +dim v|0 = dim v, where dim v|0 is the dimension of v|0 as a subspace of T0Rn, while +dim v is the dimension of v as a subspace of g. Indeed, we have the trivial inequality +dim v|0 ≤ dim v. On the other hand, if strict inequality holds, then v must contain non- +zero vector fields vanishing at zero, contradicting the fact that v ∩ h = {0}. Therefore, +since dim g1|0 = k1 and dim h1|0 = 0, we get dim v = dim v|0 = k1 as desired. +□ +Step 4: proof of the Killing property. We have so far proved the existence of i such that +g1 = i ⊕ h1, and such that any element Y ∈ i commutes with the sub-Laplacian �∆. We +now show that all such Y is a Killing vector field. +Let Y ∈ i. Since [Y, �∆] = 0, the induced flow φY +s , for s ∈ R, commutes with �∆ when +acting on smooth functions, that is, +�∆(u ◦ φY +s ) = ( �∆u) ◦ φY +s +(3.6) +for all u ∈ C∞(Rn) and s ∈ R. Recall the sub-Riemannian Hamiltonian � +H : T ∗Rn → R, +� +H(λ) = 1 +2 +L +� +i=1 +⟨λ, � +Xi⟩2, +(3.7) +for all λ ∈ T ∗Rn. By (2.13), � +H is the principal symbol of �∆. Thus, from (3.6) it follows +� +H ◦ +� +φY +s +�∗ = � +H, +for all s ∈ R, where the star denotes the pull-back, and thus +� +φY +s +�∗ is a diffeomorphism +on T ∗Rn. This means that φY +s is an isometry, as we now show. Indeed, for any given +x ∈ Rn, the restriction � +H|T ∗ +x Rn is a quadratic form on T ∗ +xRn, so (φY +s )∗ must preserve its +kernel, that is, +(φY +s )∗ ker � +H|T ∗ +φYs (x)Rn = ker � +H|T ∗ +x Rn +(3.8) + +16 +L. RIZZI AND G. STEFANI +for all x ∈ Rn. By (3.7), it holds ker � +H|T ∗ +x Rn = � +D⊥ +x , where ⊥ denotes the annihilator of a +vector space. By duality, from (3.8) we obtain that (φY +s )∗ � +Dx = � +DφYs (x) for all x ∈ Rn as +required by (1.2). Finally, for λ ∈ T ∗ +xM, let λ# ∈ Dx be uniquely defined by gx(λ#, V ) = +⟨λ, V ⟩x for all V ∈ Dx, and notice that the map λ �→ λ# is surjective on Dx. Then it holds +∥λ#∥2 +x = 2� +H(λ), see Lemma A.1. Thus, since (φY +s )∗ preserves � +H, the map (φY +s )∗ preserves +the sub-Riemannian norm, and thus g. This means that φY +s is an isometry, concluding +the proof of Theorem 1.4. +□ +3.2. Proof of Theorem 1.5. We claim that +gj = hj +for all j ≥ 2. +(3.9) +Note that (3.9) is enough to conclude the proof of Theorem 1.5, since, from (3.9) combined +with (2.11) and (2.12), we immediately get that +g = g1 ⊕ h2 ⊕ · · · ⊕ hs. +In particular, we deduce that g|0 = g1|0, which in turn implies that g must be commu- +tative, otherwise the bracket-generating condition would fail. To prove (3.9), we proceed +by induction on j ≥ 2 as follows. +Proof of the base case j = 2. We begin by proving the base case j = 2 in (3.9). To this +aim, let � +X ∈ i and �Y ∈ g1. By definition of Lie bracket, we can write +� +φ � +X +−s +� +∗ +�Y = s +�� +X, �Y +� ++ o(s) +as s → 0, +where φ � +X +s , for s ∈ R, is the flow of � +X. Since g1|x = � +D|x for all x ∈ Rn, and since � +X is +Killing (in particular (1.2) holds for its flow), we have that [� +X, �Y ]|x ∈ � +D|x for all x ∈ Rn. +Since [� +X, �Y ] ∈ g2 and so, in particular, [� +X, �Y ] is homogeneous of degree −2, we have +[� +X, �Y ]|0 = +� +j : wj=2 +aj ∂zj|0, +for some constants aj ∈ R. But we also must have that [� +X, �Y ]|0 ∈ � +D|0 and so, since +� +D|0 = span +� +∂zj : wj = 1 +� +according to Definition 2.1, [� +X, �Y ]|0 = 0, that is, [� +X, �Y ] ∈ h. We thus have proved that +[i, g1] ⊂ h2. In particular, since g1 = i ⊕ h1, we get +[i, i] ⊂ h2 +and +[i, h1] ⊂ h2, +(3.10) +from which we readily deduce (3.9) for j = 2. +Proof of the induction step. Let us assume that (3.9) holds for some j ∈ N, j ≥ 2. Since +g1 = i ⊕ h1, by the induction hypothesis we can write +gj+1 = [g1, gj] = [g1, hj] = [i, hj] + [h1, hj] = [i, hj] + hj+1. +We thus just need to show that [i, hj] ⊂ hj+1 for all j ∈ N with j ≥ 2. Note that we +actually already proved the case j = 1 in (3.10). Again arguing by induction (taking +j = 1 as base case), by the Jacobi identity and (3.10) we have +[i, hj+1] = [i, [h1, hj]] = [h1, [hj, i]] + [hj, [i, h1]] ⊂ [h1, hj+1] + [hj, h2] = hj+2 + +FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS +17 +as desired, concluding the proof of the induction step. +□ +Remark 3.2 (Proof of Theorem 1.5 in the case h = {0}). The proof of Theorem 1.5 is +much simpler if the nilpotent approximation (Rn, � +F) is a Carnot group, i.e., h = {0}. +Indeed, in this case, the base case j = 2 in (3.9) immediately implies that g2 = h2 = {0}, +which in turn gives g = g1, so that g is commutative. +3.3. Proof of Theorem 1.8. In the following, we assume that the reader is familiar with +the notions of upper gradient and of q-upper gradient, see [5] for the precise definitions. +The next two lemmas are proved in [27] for sub-Riemannians structures on Rn equipped +with the Lebesgue measure, and are immediately extended to the weighted case. +Lemma 3.3. Let (M, d, m) be as in Theorem 1.8. If u ∈ C(M) and 0 ≤ g ∈ L1 +loc(M, m) +be an upper gradient of u, then u ∈ HW1,1 +loc(M, m) with ∥∇u∥ ≤ g m-a.e. In particular, if +u ∈ Lip(M, d), then ∥∇u∥ ≤ Lip(u). +Proof. Without loss of generality we may assume that M = Ω ⊂ Rn is a bounded open +set, the sub-Riemannian structure is induced by a family of smooth bracket-generating +vector fields F = {X1, . . ., XL} on Ω and m = θL n, where θ: Ω → [0, ∞) is smooth +and satisfies 0 < infΩ θ ≤ supΩ θ < ∞. Hence, L1(Ω, θL n) = L1(Ω, L n) as sets, with +equivalent norms, so that 0 ≤ g ∈ L1 +loc(Ω, L n) is an upper gradient of u ∈ C(Ω). Hence, +by [27, Th. 11.7], we get that u ∈ HW1,1 +loc(Ω, L n), with ∥∇u∥ ≤ g L n-a.e., and thus +θL n-a.e., on Ω. By definition of distributional derivative, we can write +� +Ω v Xiu dx = +� +Ω u [−Xiv + div(Xi)v] dx, +∀ v ∈ C1 +c (Ω), i = 1, . . . , L, +where div denotes the Euclidean divergence. +We apply the above formula with test +function v = θw, for any w ∈ C1 +c (Ω), getting +� +Ω w Xiu θ dx = +� +Ω u +� +−Xiw + div(Xi)w + Xiθ +θ w +� +θ dx, +∀ w ∈ C1 +c (Ω), i = 1, . . . , L. +The function within square brackets is the adjoint X∗ +i w with respect to the measure +θL n. It follows that HW1,q(Ω, θL n) = HW1,q(Ω, L n) as sets, with equivalent norms. In +particular, u ∈ W1,1 +D,loc(Ω, θL n) as desired. +□ +Lemma 3.4 (Meyers–Serrin). Let (M, d, m) be as in Theorem 1.8 and let q ∈ [1, ∞). +Then HW1,q(M, m) ∩ C∞(M) is dense in HW1,q(M, m). +Proof. Up to a partition of unity and exhaustion argument, we can reduce to the case +M = Ω ⊂ Rn is a bounded open set and m = θL n, where θ: Ω → [0, ∞) is as in the +previous proof, so that HW1,q(Ω, L n) = HW1,q(Ω, θL n) as sets, with equivalent norms. +In particular, we can assume that θ ≡ 1. This case is proved in [27, Th. 11.9]. +□ +Lemma 3.5. Let (M, d, m) be as in Theorem 1.8 and let q ∈ [1, ∞). If u ∈ HW1,q(M, m), +then ∥∇u∥ is the minimal q-upper gradient of u. +Proof. Let us first prove that ∥∇u∥ is a q-upper gradient of u. Indeed, by Lemma 3.4, we +can find (uk)k∈N ⊂ HW1,q(M, m) ∩ C∞(M) such that uk → u in HW1,q(M, m) as k → ∞. + +18 +L. RIZZI AND G. STEFANI +It is well-known that the sub-Riemannian norm of the gradient of a smooth function is +an upper gradient, see [27, Prop. 11.6]. Thus, for uk it holds +|uk(γ(1)) − uk(γ(0))| ≤ +� +γ ∥∇uk∥ ds. +Arguing as in [28, p. 179], using Fuglede’s lemma (see [28, Lem. 7.5 and Sec. 10]), we pass +to the limit for k → ∞ in the previous equality, outside a q-exceptional family of curves. +This proves that any Borel representative of ∥∇u∥ is a q-upper gradient of u. +We now prove that ∥∇u∥ is indeed minimal. Let 0 ≤ g ∈ Lq(M, m) be any q-upper +gradient of u. Arguing as in [28, p. 194], we can find a sequence (gk)k∈N ⊂ Lq(M, m) of +upper gradients of u such that gk ≥ g for all k ∈ N and gk → g both pointwise m-a.e. +on M and in Lq(M, m) as k → ∞. By Lemma 3.3, we thus must have that ∥∇u∥ ≤ gk +m-a.e. on M for all k ∈ N. Hence, passing to the limit, we conclude that ∥∇u∥ ≤ g m-a.e. +on M for any q-upper gradient g, concluding the proof. +□ +We are now ready to deal with the proof of Theorem 1.8. +Proof of (i). Recall that, here, q > 1. We begin by claiming that +W1,q(M, d, m) ⊂ HW1,q(M, m) +(3.11) +isometrically, with ∥∇u∥ = |Du|w,q. Indeed, let u ∈ W1,q(M, d, m). By a well-known +approximation argument, combining [5, Prop. 4.3, Th. 5.3 and Th. 7.4], we find (uk)k∈N ⊂ +Lip(M, d) ∩ W1,q(M, d, m) such that +uk → u +and +|Duk|w,q → |Du|w,q +in Lq(M, m). +(3.12) +Since uk ∈ Lip(M, d), by Lemma 3.3 we know that uk ∈ HW1,q(M, m). +Hence, by +Lemma 3.5, |Duk|w,q = ∥∇uk∥, and we immediately get that +sup +k∈N +� +M ∥∇uk∥q dm < ∞. +Therefore, up to passing to a subsequence, (Xiuk)k∈N is weakly convergent in Lq(M, m), +say Xiuk ⇀ αi ∈ Lq(M, m), for all i = 1, . . ., L. We thus get that u ∈ HW1,q(M, m) with +Xiu = αi and thus ∇u = +�L +i=1 αiXi. By stability of q-upper gradients, [5, Th. 5.3 and +Thm. 7.4], ∥∇u∥ is a q-upper gradient of u. By semi-continuity of the norm, we obtain +� +M ∥∇u∥q dm ≤ lim inf +k→∞ +� +M ∥∇uk∥q dm = +� +M |Du|q +w,q dm, +where we used (3.12). By definition of minimal q-upper gradient we thus get that ∥∇u∥ = +|Du|w,q m-a.e., and the claimed inclusion in (3.11) immediately follows. +We now observe that it also holds +HW1,q(M, m) ∩ C∞(M) ⊂ W1,q(M, d, m), +(3.13) +with ∥∇u∥ = |Du|w,q. We just need to notice that, if u ∈ C∞(M), then ∥∇u∥ is an +upper gradient of u, see [27, Prop. 11.6]. Therefore, by Lemma 3.3, ∥∇u∥ must coincide +with the minimal q-upper gradient of u, i.e., ∥∇u∥ = |Du|w m-a.e., and (3.13) readily +follows. In view of the isometric inclusions (3.11) and (3.13), and of the density provided +by Lemma 3.4, this concludes the proof of (i). +□ + +FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS +19 +Proof of (ii). Let us assume that (M, d, m) satisfies the CD(K, ∞) property for some +K ∈ R. +By the previous point (i), we know that (M, d, m) satisfies the RCD(K, ∞) +property. Consequently, since clearly C∞ +c (M) ⊂ W1,2(M, d, m) by (3.13), [6, Rem. 6.3] +(even if the measure m is σ-finite, see [4, Sec. 7] for a discussion) implies that +1 +2 +� +M ∆v ∥∇u∥2 dm − +� +M v g(∇u, ∇∆u) dm ≥ K +� +M v ∥∇u∥2 dm +for all u, v ∈ C∞ +c (M) with v ≥ 0 on M, from which we readily deduce (1.1). +□ +Remark 3.6. The above proofs work for more general measures m. Namely, we can +assume that, locally on any bounded coordinate neighborhood Ω ⊂ Rn, m = θL n with +θ ∈ W1,1(Ω, L n) ∩ L∞(Ω, L n). +In this case, the positivity of m corresponds to the +requirement that θ is locally essentially bounded from below away from zero, in charts. +3.4. Proof of Theorem 1.10. We prove the two points in the statement separately. +Proof of (i). The case p = 0 has been already considered by Juillet in [32]. For p > 0, +we can argue as follows. Let A0 = [−ℓ −1, −ℓ]×[0, 1] and A1 = [ℓ, ℓ + 1]×[0, 1] for ℓ > 0. +We will shortly prove that the midpoint set +A1/2 = +� +q ∈ R2 : ∃ q0 ∈ A0, ∃ q1 ∈ A1 with d(q, q0) = d(q, q1) = 1 +2 d(q0, q1) +� +satisfies +A1/2 ⊂ [−1 − εℓ, 1 + εℓ] × [0, 1] +(3.14) +for some εℓ > 0, with εℓ ↓ 0 as ℓ → ∞. Since mp(A0) = mp(A1) ∼ ℓp as ℓ → ∞, we get +� +mp(A0) mp(A1) > mp(A1/2) +for large ℓ > 0. This contradicts the logarithmic Brunn–Minkowski BM(0, ∞) inequality, +which is a consequence of the CD(0, ∞) condition, see [50, Th. 30.7]. +To prove (3.14), let qi ∈ Ai, qi = (xi, yi), and let γ(t) = (x(t), y(t)), t ∈ [0, 1], be a +geodesic such that γ(i) = qi, with i = 0, 1. We first note that +min{y0, y1} ≤ y(t) ≤ max{y0, y1} +for all t ∈ [0, 1], +(3.15) +since any curve that violates (3.15) can be replaced by a strictly shorter one satisfy- +ing (3.15). In particular, we get that A1/2 ⊂ R × [0, 1]. Let us now observe that +|xa − xb| ≤ d(a, b) ≤ |xa − xb| + +|ya − yb| +max{|xa|, |xb|} +for all a = (xa, ya) and b = (xb, yb) with xa, xb ̸= 0. Therefore, if q = (x, y) ∈ A1/2, then +|x − x0| ≤ d(q, q0) = 1 +2 d(q0, q1) ≤ ℓ + 1 + O(1/ℓ) +and, similarly, |x − x1| ≤ ℓ + 1 + O(1/ℓ). Since x0 ∈ [−ℓ − 1, −ℓ] and x1 ∈ [ℓ, ℓ + 1], we +deduce that |x| ≤ 1 + O(1/ℓ), concluding the proof of the claimed (3.14). +□ + +20 +L. RIZZI AND G. STEFANI +Proof of (ii). Out of the negligible set {x = 0}, the metric g on Gp given by (1.5) is +locally Riemannian. Recalling (1.6) and (1.7), the BE(K, ∞) inequality (1.1) is implied by +the lower bound Ric∞,V ≥ K via Bochner’s formula, where Ric∞,V is the ∞-Bakry–Émery +Ricci tensor of (R2, g, e−V volg), see [50, Ch. 14, Eqs. (14.36) – (14.51)]. By Lemma 3.7 +below, we have Ric∞,V ≥ 0 for all p ≥ 1, concluding the proof. +□ +Lemma 3.7. Let p ∈ R and N > 2. The N-Bakry–Émery Ricci tensor of the Grushin +metric (1.5), with weighted measure mp = |x|p dx dy, for all x ̸= 0 is +RicN,V = p − 1 +x2 +g −(p + 1)2 +N − 2 +dx ⊗ dx +x2 +, +with the convention that 1/∞ = 0. +Proof. The N-Bakry–Émery Ricci tensor of a n-dimensional weighted Riemannian struc- +ture (g, e−V volg), for N > n, is given by +RicN,V = Ricg + HessgV − dV ⊗ dV +N − n , +(3.16) +see [50, Eq. (14.36)]. In terms of the frame (1.4), the Levi-Civita connection is given by +∇XX = ∇XY = 0, +∇Y X = −1 +xY, +∇Y Y = 1 +xX, +whenever x ̸= 0. Recalling that, from (1.7), V (x) = −(p + 1) log |x|, for x ̸= 0, we obtain +Ricg = − 2 +x2 g, +HessgV = (p + 1) +x2 +g, +dV = −p + 1 +x +dx, +(3.17) +whenever x ̸= 0. The conclusion thus follows by inserting (3.17) into (3.16). +□ +3.5. Proof of Theorem 1.11. The statement is a consequence of the geodesic convexity +of G+ +p and the computation of the N-Bakry–Émery curvature in Lemma 3.7. Since the +proof uses quite standard arguments, we simply sketch its main steps. +The interior of G+ +p , i.e., the open half-plane, can be regarded as a (non-complete) +weighted Riemannian manifold with metric g as in (1.5) and weighted volume as in (1.7). +Let µ0, µ1 ∈ P2(G+ +p ), µ0, µ1 ≪ mp, with bounded support contained in the Riemannian +region {x > ε}, for some ε ≥ 0. +Let (µs)s∈[0,1] be a W2-geodesic joining µ0 and µ1. +By a well-known representation +theorem (see [50, Cor. 7.22]), there exists ν ∈ P(Geo(G+ +p )), supported on the set +Γ = (e0 × e1)−1(supp µ0 × supp µ1), such that µs = (es)♯ν for all s ∈ [0, 1]. Since the +set {x ≥ ε} is a geodesically convex subset of the full Grushin plane Gp (by the same +argument of [46, Prop. 5]), any γ ∈ Γ is contained for all times in the region {x > 0}. +Therefore, Γ is a set of Riemannian geodesics contained in the weighted Riemannian struc- +ture ({x > 0}, g, e−V volg). By Lemma 3.7, we have RicN,V ≥ 0 for all N ≥ Np, where Np +is as in (1.8). At this point, a standard argument shows that the Rényi entropy is convex +along Wasserstein geodesics joining µ0 with µ1, see the proof of [49, Th. 1.7] for example. +The extension to µ0, µ1 ∈ P2(G+ +p ), with µ0, µ1 ≪ mp and compact support possibly +touching the singular region {x = 0}, is achieved via a standard approximation argument. +More precisely, one reduces to the previous case and exploits the stability of optimal +transport [50, Th. 28.9] and the lower semi-continuity of the Rényi entropy [50, Th. 29.20]. + +FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS +21 +Finally, the extension to general µ0, µ1 ∈ P2(G+ +p ) follows the routine argument outlined +in [9, Rem. 2.12], which works when µs = (es)♯ν, s ∈ [0, 1], and ν is concentrated on a set +of non-branching geodesics. This proves the ‘if’ part of the statement. +The ‘only if’ part is also standard. The CD(0, N) condition for N > 2 implies that, on +the Riemannian region {x > 0}, RicN,V ≥ 0, but this is false for N < Np. +The fact that G+ +p is infinitesimally Hilbertian follows from Remark 1.9, by noting that +mp is positive and smooth out of the closed set {x = 0}, which has zero measure. An +alternative proof follows from the observation that G+ +p is a Ricci limit, see [42]. +□ +Appendix A. Gradient and Laplacian representations formulas +For the reader’s convenience, in this appendix we provide a short proof of the repre- +sentation formulas (2.5) and (2.7), in the rank-varying case. +Lemma A.1. For λ ∈ T ∗M, let λ# ∈ D be uniquely defined by +g(λ#, V ) = ⟨λ, V ⟩ +for all V ∈ D, where ⟨·, ·⟩ denotes the action of covectors on vectors. Then +∥λ#∥2 = +L +� +i=1 +� +λ#, Xi +�2. +(A.1) +As a consequence, if λ, µ ∈ T ∗M, then +g(λ#, µ#) = +L +� +i=1 +⟨λ, Xi⟩⟨µ, Xi⟩. +(A.2) +Proof. Given u ∈ RL, we set Xu = �L +i=1 uiXi and define +u∗ ∈ argmin +� +v �→ |v| : v ∈ RL, Xv = Xu +� +. +In other words, for Xu ∈ D, u∗ is the element of minimal Euclidean norm such that +Xu∗ = Xu. Note that, by definition, it holds ∥Xu∥ = |u∗|. We thus have +∥λ#∥ = sup +� +g(λ#, X) : ∥X∥ = 1, X ∈ D +� += sup +� +g(λ#, Xu) : |u∗| = 1, u ∈ RL� +. +We now claim that +sup +� +g(λ#, Xu) : |u∗| = 1, u ∈ RL� += sup +� +g(λ#, Xu) : |u| = 1, u ∈ RL� +. +(A.3) +Indeed, the inequality ≤ in (A.3) is obtained by observing that Xu = Xu∗ for any u ∈ RL. +To prove the inequality ≥ in (A.3), we observe that, if u ∈ RL is such that |u| = 1 and +0 < |u∗| < 1, then v = u/|u∗| satisfies |v∗| = 1 and gives +g(λ#, Xv) > g(λ#, Xv) |u∗| = g(λ#, Xu). +(A.4) +Furthermore, if |u| = 1 and u∗ = 0, then Xu = 0 so also in this case we find v ∈ Rn with +v∗ = 1 such that (A.4) holds. This ends the proof of the claimed (A.3). Hence, since +g(λ#, Xu) = +L +� +i=1 +g(λ#, Xi) ui, + +22 +L. RIZZI AND G. STEFANI +we easily conclude that +∥λ#∥ = sup +� +g(λ#, Xu) : |u| = 1, u ∈ RL� += +� +� +� +� +L +� +i=1 +g(λ#, Xi)2, +proving (A.1). Equality (A.2) then follows by polarization. +□ +Corollary A.2. The following formulas hold: +∇u = +L +� +i=1 +Xiu Xi, +(A.5) +∆u = +L +� +i=1 +� +X2 +i u + Xiu divm(Xi) +� +, +(A.6) +g(∇u, ∇v) = +L +� +i=1 +Xiu Xiv, +(A.7) +for all u, v ∈ C∞(M). In particular, ∥∇u∥ = +�L +i=1(Xiu)2 for all u ∈ C∞(M). +Proof. We prove each formula separately. +Proof of (A.5). Recalling the definition in (2.4), we can pick λ = du in (A.2) to get +� +du, µ#� += g(∇u, µ#) = +L +� +i=1 +⟨du, Xi⟩⟨µ, Xi⟩ += +L +� +i=1 +Xiu ⟨µ, Xi⟩ = g +� +µ#, +L +� +i=1 +Xiu Xi +� +whenever µ ∈ T ∗ +xM. Since the map #: T ∗ +xM → Dx is surjective, we immediately get (A.5). +Proof of (A.6). Recall that +divm(fX) = Xf + f divm(X) +for any f ∈ C∞(M) and X ∈ Γ(TM). Hence, from the definition in (2.6), we can compute +∆u = divm(∇u) = +L +� +i=1 +divm(Xiu Xi) = +L +� +i=1 +� +X2 +i u Xi + Xiu divm(Xi) +� +, +which is the desired (A.6). +Proof of (A.7). Choosing λ = du and µ = dv in (A.2), we can compute +g(∇u, ∇v) = +L +� +i=1 +⟨du, Xi⟩ ⟨dv, Xi⟩ = +L +� +i=1 +Xiu Xiv +and the proof is complete. +□ + +FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS +23 +References +[1] A. Agrachev, D. Barilari, and L. Rizzi, Curvature: a variational approach, Mem. Amer. Math. Soc. +256 (2018), no. 1225, v+142. +[2] A. Agrachev, D. Barilari, and U. Boscain, A comprehensive introduction to sub-Riemannian geome- +try, Cambridge Studies in Advanced Mathematics, vol. 181, Cambridge University Press, Cambridge, +2020. From the Hamiltonian viewpoint, With an appendix by Igor Zelenko. +[3] L. Ambrosio, Calculus, heat flow and curvature-dimension bounds in metric measure spaces, Pro- +ceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. I. 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I, Acta Math. 196 (2006), no. 1, 65–131. +[49] +, On the geometry of metric measure spaces. II, Acta Math. 196 (2006), no. 1, 133–177. +[50] C. Villani, Optimal transport, Grundlehren der mathematischen Wissenschaften [Fundamental Prin- +ciples of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009. Old and new. + +FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS +25 +(L. Rizzi) Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265, +34136 Trieste (TS), Italy +Email address: lrizzi@sissa.it +(G. Stefani) Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265, +34136 Trieste (TS), Italy +Email address: gstefani@sissa.it or giorgio.stefani.math@gmail.com + diff --git a/1tAyT4oBgHgl3EQf1flH/content/tmp_files/load_file.txt b/1tAyT4oBgHgl3EQf1flH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3513f8adfcba088b5ced0511fd9bc4cf17594caf --- /dev/null +++ b/1tAyT4oBgHgl3EQf1flH/content/tmp_files/load_file.txt @@ -0,0 +1,1271 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf,len=1270 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='00735v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='DG] 2 Jan 2023 FAILURE OF CURVATURE-DIMENSION CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS VIA TANGENT ISOMETRIES LUCA RIZZI AND GIORGIO STEFANI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We prove that, on any sub-Riemannian manifold endowed with a positive smooth measure, the Bakry–Émery inequality for the corresponding sub-Laplacian, 1 2∆(∥∇u∥2) ≥ g(∇u, ∇∆u) + K∥∇u∥2, K ∈ R, implies the existence of enough Killing vector fields on the tangent cone to force the latter to be Euclidean at each point, yielding the failure of the curvature-dimension condition in full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Our approach does not apply to non-strictly-positive measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In fact, we prove that the weighted Grushin plane does not satisfy any curvature-dimension condition, but, nevertheless, does admit an a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' pointwise version of the Bakry–Émery inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' As recently observed by Pan and Montgomery, one half of the weighted Grushin plane satisfies the RCD(0, N) condition, yielding a counterexample to gluing theorems in the RCD setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Introduction and statements In the last twenty years, there has been an impressive effort in extending the concept of ‘Ricci curvature lower bound’ to non-Riemannian structures, and even to general metric spaces equipped with a measure (metric-measure spaces, for short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We refer the reader to the ICM notes [3] for a survey of this line of research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' There are two distinct points of view on the matter, traditionally known as the La- grangian and Eulerian approaches, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The Lagrangian point of view is the one adopted by Lott–Villani and Sturm [36, 48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In this formulation, Ricci curvature lower bounds are encoded by convexity-type inequalities for entropy functionals on the Wasserstein space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Such inequalities are called curvature-dimension conditions, CD(K, N) for short, where K ∈ R represents the lower bound on the curvature and N ∈ [1, ∞] stands for an upper bound on the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The Eulerian point of view, instead, employs the metric-measure structure to define an energy form and, in turn, an associated diffusion operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The notion of Ricci curvature lower bound is therefore encoded in the so-called Bakry–Émery inequality, BE(K, N) for short, for the diffusion operator, which can be expressed in terms of a suitable Gamma calculus, see the monograph [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Thanks to several key contributions [4, 6, 7, 24], the Lagrangian and the Eulerian ap- proaches are now known to be essentially equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, CD(K, N) always Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Primary 53C17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Secondary 54E45, 28A75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Sub-Riemannian manifold, CD(K, ∞) condition, Bakry–Émery inequality, infinitesimally Hilbertian, Grushin plane, privileged coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 1 2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' RIZZI AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' STEFANI implies BE(K, N) in infinitesimal Hilbertian metric-measure spaces, as introduced in [25], while the converse implication requires further technical assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Such synthetic theory of curvature-dimension conditions, besides being consistent with the classical notions of Ricci curvature and dimension on smooth Riemannian manifolds, is stable under pointed-measure Gromov–Hausdorff convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Furthermore, it yields a comprehensive approach for establishing all results typically associated with Ricci cur- vature lower bounds, like Poincaré, Sobolev, log-Sobolev and Gaussian isoperimetric in- equalities, as well as Brunn–Minkowski, Bishop–Gromov and Bonnet–Myers inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The sub-Riemannian framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Although the aforementioned synthetic cur- vature-dimension conditions embed a large variety of metric-measure spaces, a relevant and widely-studied class of smooth structures is left out—the family of sub-Riemmanian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' A sub-Riemannian structure is a natural generalization of a Riemannian one, in the sense that its distance is induced by a scalar product that is defined only on a smooth sub-bundle of the tangent bundle, whose rank possibly varies along the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' See the monographs [2,40,45] for a detailed presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The first result in this direction was obtained by Driver–Melcher [23], who proved that an integrated version of the BE(K, ∞), the so-called pointwise gradient estimate for the heat flow, is false for the three-dimensional Heisenberg group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In [31], Juillet proved the failure of the CD(K, ∞) property for all Heisenberg groups (and even for the strictly related Grushin plane, see [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Later, Juillet [33] extended his result to any sub-Riemannian manifold endowed with a possibly rank-varying distribution of rank strictly smaller than the manifold’s dimension, and with any positive smooth measure, by exploiting the notion of ample curves introduced in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The idea of [31,33] is to construct a counterexample to the Brunn–Minkowski inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The ‘no-CD theorem’ of [31] was extended to all Carnot groups by Ambrosio and the second-named author in [8, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6] with a completely different technique, namely, by exploiting the optimal version of the reverse Poincaré inequality obtained in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In the case of sub-Riemannian manifolds endowed with an equiregular distribution and a positive smooth measure, Huang–Sun [29] proved the failure of the CD(K, N) condition for all values of K ∈ R and N ∈ (1, ∞) contradicting a bi-Lipschitz embedding result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Very recently, in order to address the structures left out in [33], Magnabosco–Rossi [37] recently extended the ‘no-CD theorem’ to almost-Riemannian manifolds M of dimension 2 or strongly regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The approach of [37] relies on the localization technique developed by Cavalletti–Mondino [19] in metric-measure spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' To complete the picture, we mention that several replacements for the Lott–Sturm– Villani curvature-dimension property have been proposed and studied in the sub-Rieman- nian framework in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Far from being complete, we refer the reader to [11–15,38] for an account on the Lagrangian approach, to [17] concerning the Eulerian one, and finally to [47] for a first link between entropic inequalities and contraction properties of the heat flow in the special setting of metric-measure groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Main aim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' At the present stage, a ‘no-CD theorem’ for sub-Riemannian structures in full generality is missing, since the aforementioned approaches [8,23,29,31,33,37] either require the ambient space to satisfy some structural assumptions, or leave out the infinite dimensional case N = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS 3 The main aim of the present paper is to fill this gap by showing that (possibly rank- varying) sub-Riemannian manifolds do not satisfy any curvature bound in the sense of Lott–Sturm–Villani or Bakry–Émery when equipped with a positive smooth measure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', a Radon measure whose density in local charts with respect to the Lebesgue measure is a strictly positive smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Failure of the Bakry–Émery inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The starting point of our strategy is the weakest curvature-dimension condition, as we now define.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1 (Bakry–Émery inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We say that a sub-Riemannian manifold (M, d) endowed with a positive smooth measure m satisfies the Bakry–Émery BE(K, ∞) inequality, for K ∈ R, if 1 2 ∆(∥∇u∥2) ≥ g(∇u, ∇∆u) + K∥∇u∥2 for all u ∈ C∞(M), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1) where ∆ is the corresponding sub-Laplacian, and ∇ the sub-Riemannian gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Our first main result is the following rigidity property for sub-Riemannian structures supporting the Bakry–Émery inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2 (no-BE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let (M, d) be a complete sub-Riemannian manifold endowed with a positive smooth measure m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' If (M, d, m) satisfies the BE(K, ∞) inequality for some K ∈ R, then rank Dx = dim M at each x ∈ M, so that (M, d) is Riemannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The idea behind our proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2 is to show that the metric tangent cone in the sense of Gromov [26] at each point of (M, d) is Euclidean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' This line of thought is somehow reminiscent of the deep structural result for RCD(K, N) spaces, with K ∈ R and N ∈ (1, ∞), proved by Mondino–Naber [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' However, differently from [39], Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2 provides information about the metric tangent cone at each point of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Showing that the distribution D is Riemannian at almost every point in fact would not be enough, as this would not rule out almost-Riemannian structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Starting from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1), we first blow-up the sub-Riemannian structure and pass to its metric-measure tangent cone, showing that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1) is preserved with K = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Note that, in this blow-up procedure, the positivity of the density of m is crucial, since otherwise the resulting metric tangent cone would be endowed with the null measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The resulting blown-up sub-Riemannian space is isometric to a homogeneous space of the form G/H, where G = exp g is the Carnot group associated to the underlying (finite-dimensional and stratified) Lie algebra g of bracket-generating vector fields, and H = exp h is its subgroup corresponding to the Lie subalgebra h of vector fields vanishing at the origin, see [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Of course, the most difficult case is when H is non-trivial, that is, the tangent cone is not a Carnot group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' At this point, the key idea is to show that the Bakry–Émery inequality BE(K, ∞) implies the existence of special isometries on the tangent cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3 (Sub-Riemannian isometries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let M be a sub-Riemannian manifold, with distribution D and metric g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' A diffeomorphism φ : M → M is an isometry if (φ∗D)|x = Dφ(x) for all x ∈ M, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2) and, furthermore, φ∗ is an orthogonal map with respect to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We say that a smooth vector field V is Killing if its flow φV t is an isometry for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 4 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' RIZZI AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' STEFANI For precise definitions of g and h in the next statement, we refer to Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4 (Existence of Killing fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let (M, d) be a complete sub-Riemannian manifold equipped with a positive smooth measure m If (M, d, m) satisfies the BE(K, ∞) inequality for some K ∈ R, then, for the nilpotent approximation at any given point, there exists a vector space i ⊂ g1 such that g1 = i ⊕ h1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3) and every Y ∈ i is a Killing vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The existence of the space of isometries i forces the Lie algebra g to be commutative and of maximal rank, thus implying that the original manifold (M, d) was in fact Riemannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5 (Killing implies commutativity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' If there exists a subspace i ⊂ g1 of Killing vector fields such that g1 = i ⊕ h1, then g is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5 states that, if a Carnot group contains enough horizontal symmetries, then it must be commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' As it will be evident from its proof, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5 holds simply assuming that, for each V ∈ i, the flow φV t is pointwise distribution-preserving, namely it satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2), without being necessarily isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Infinitesimal Hilbertianity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The Bakry–Émery inequality BE(K, ∞) in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1) is a consequence of the CD(K, ∞) condition as soon as the ambient metric-measure space is infinitesimal Hilbertian as defined in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let (X, d) be a complete separable metric space, m be a locally bounded Borel mea- sure, and q ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We let |Du|w,q ∈ Lq(X, m) be the minimal q-upper gradient of a measurable function u : X → R, see [5, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We define the Banach space W1,q(X, d, m) = {u ∈ Lq(X, m) : |Du|w,q ∈ Lq(X, m)} with the norm ∥u∥W1,q(X,d,m) = � ∥u∥q Lq(X,m) + ∥|Du|w,q∥q Lq(X,m) �1/q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6 (Infinitesimal Hilbertianity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' A metric measure space (X, d, m) is in- finitesimally Hilbertian if W1,2(X, d, m) is a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The infinitesimal Hilbertianity of sub-Riemannian structures has been recently proved in [35], with respect to any Radon measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2 immediately yields the following ‘no-CD theorem’ for sub-Riemannian manifolds, thus extending all the aforementioned results [8,23,29,31,33,37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7 (no-CD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let (M, d) be a complete sub-Riemannian manifold endowed with a positive smooth measure m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' If (M, d, m) satisfies the CD(K, ∞) condition for some K ∈ R, then (M, d) is Riemannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' However, since the measure in Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7 is positive and smooth, we can avoid to rely on the general result of [35], instead providing a simpler and self-contained proof of the infinitesimal Hilbertianity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, we prove the following result, which actually refines [35, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6] in the case of smooth measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In the following, HW1,q(M, m) denotes the sub-Riemannian Sobolev spaces (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS 5 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='8 (Infinitesimal Hilbertianity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let q ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let (M, d) be a complete sub- Riemannian manifold equipped with a positive smooth measure m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (i) W1,q(M, d, m) = HW1,q(M, m), with |Du|w,q = ∥∇u∥ m-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' on M for all u ∈ W1,q(M, d, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, taking q = 2, (M, d, m) is infinitesimally Hilbertian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (ii) If (M, d, m) satisfies the CD(K, ∞) condition for some K ∈ R, then the Bakry– Émery BE(K, ∞) inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1) holds on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Note that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='8 holds for less regular measures, see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='9 (The case of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' smooth measures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='8 can be adapted also to the case of a Borel and locally finite measure m which is smooth and positive only on Ω, where Ω ⊂ M is an open set with m(∂Ω) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In this case, we obtain HW1,q(Ω, m) = W1,q(Ω, d, m), with |Du|w,q = ∥∇u∥ m-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' on Ω for all u ∈ W1,q(Ω, d, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, if m is smooth and positive out of a closed set Z, with m(Z) = 0, an elementary ap- proximation argument proves that (M, d, m) is infinitesimally Hilbertian and, if (M, d, m) satisfies the CD(K, ∞) condition for K ∈ R, then the Bakry-Émery BE(K, ∞) inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1) holds on M \\Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' This is the case, for example, of the Grushin planes and half-planes with weighted measures of Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The proof follows the same argument of the one of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='8, exploiting the locality of the q-upper gradient, see for example [5, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2] and [25, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6], and similar properties for the distributional derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' An alternative approach to the ‘no-CD theorem’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We mention an alternative proof of the ‘no-CD theorem’ for almost-Riemannian structures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', sub-Riemannian structures that are Riemannian outside a closed nowhere dense singular set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The strategy relies on the Gromov-Hausdorff continuity of the metric tangent at interior points of geodesics in RCD(K, N) spaces, with N < ∞, proved by Deng in [22], For example, consider the standard Grushin plane (introduced in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5) equipped with a smooth positive measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The curve γ(t) = (t, 0), t ∈ R, is a geodesic between any two of its point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The metric tangent at γ(t) is (isometric to) the Euclidean plane for every t ̸= 0, while it is (isometric to) the Grushin plane itself for t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Since the Grushin plane cannot be bi-Lipschitz embedded into the Euclidean plane, the two spaces are at positive Gromov-Hausdorff distance, contradicting the continuity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' This strategy has a few drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' On the one hand, it relies on the (non-trivial) machinery developed in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Consequently, this argument does not work in the case N = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' On the other hand, the formalization of this strategy for general almost-Rie- mannian structures requires certain quantitative bi-Lipschitz non-embedding results for almost-Riemannian structures into Euclidean spaces, which we are able to prove only under the same assumptions of [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Weighted Grushin structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' When the density of the smooth measure is al- lowed to vanish, the ‘no-CD theorem’ breaks down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In fact, in this situation, the following two interesting phenomena occur: (A) the Bakry-Émery BE(K, ∞) inequality no longer implies the CD(K, ∞) condition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (B) there exist almost-Riemannian structures with boundary satisfying the CD(0, N) condition for N ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 6 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' RIZZI AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' STEFANI We provide examples of both phenomena on the so-called weighted Grushin plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' This is the sub-Riemannian structure on R2 induced by the family F = {X, Y }, where X = ∂x, Y = x ∂y, (x, y) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4) The induced distribution D = span{X, Y } has maximal rank outside the singular region S = {x = 0} and rank 1 on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Since [X, Y ] = ∂y on R2, the resulting sub-Riemannian metric space (R2, d) is Polish and geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' It is almost-Riemannian in the sense that, out of S, the metric is locally equivalent to the Riemannian one given by the metric tensor g = dx ⊗ dx + 1 x2 dy ⊗ dy, x ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5) We endow the metric space (R2, d) with the weighted Lebesgue measure mp = |x|p dx dy, where p ∈ R is a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The choice p = −1 corresponds to the Riemannian density volg = 1 |x| dx dy, x ̸= 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6) so that mp = e−V volg, V (x) = −(p + 1) log |x|, x ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7) We call the metric-measure space Gp = (R2, d, mp) the (p-)weighted Grushin plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We can now state the following result, illustrating phenomenon (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let p ∈ R and let Gp = (R2, d, mp) be the weighted Grushin plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (i) If p ≥ 0, then Gp does not satisfy the CD(K, ∞) property for all K ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (ii) If p ≥ 1, then Gp satisfies the BE(0, ∞) inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1) almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' To prove (i), we show that the corresponding Brunn–Minkowski inequality is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In fact, the case p = 0 is due to Juillet [32], while the case p > 0 can be achieved via a simple argument which was pointed out to us by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Pan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Claim (ii), instead, is obtained by direct computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Somewhat surprisingly, the weighted Grushin half -plane G+ p —obtained by restricting the metric-measure structure of Gp to the (closed) half-plane [0, ∞)×R—does satisfy the CD(0, N) condition for sufficiently large N ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Precisely, we can prove the following result, illustrating phenomenon (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let p ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The weighted Grushin half-plane G+ p satisfies the CD(0, N) condition if and only if N ≥ Np, where Np ∈ (2, ∞] is given by Np = (p + 1)2 p − 1 + 2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='8) with the convention that N1 = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Furthermore, G+ p is infinitesimally Hilbertian, and it is thus an RCD(0, N) space for N ≥ Np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' While we were completing this work, Pan and Montgomery [41] observed that the spaces built in [20, 42] as Ricci limits are actually the weighted Grushin half-spaces presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Our construction and method of proof are more direct with respect to the approach of [20,42], and easily yield sharp dimensional bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Counterexample to gluing theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We end this introduction with an inter- esting by-product of our analysis, in in connection with the so-called gluing theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Perelman’s Doubling Theorem [43, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2] states that a finite dimensional Alexan- drov space with a curvature lower bound can be doubled along its boundary yielding an Alexandrov space with same curvature lower bound and dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' This result has been extended by Petrunin [44, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1] to the gluing of Alexandrov spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' It is interesting to understand whether these classical results hold true for general metric-measure spaces satisfying synthetic Ricci curvature lower bounds in the sense of Lott–Sturm–Villani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In [34], the gluing theorem was proved for CD(K, N) spaces with Alexandrov curvature bounded from below (while it is false for MCP spaces, see [46]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Here we obtain that, in general, the assumption of Alexandrov curvature bounded from below cannot be removed from the results in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' More precisely, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='10 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='11, and the fact that the metric-measure double of the Grushin half-plane G+ p is Gp (see [46, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 6]) yield the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='12 (Counterexample to gluing in RCD spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' For all N ≥ 10, there exists a geodesically convex RCD(0, N) metric-measure space with boundary such that its metric- measure double does not satisfy the CD(K, ∞) condition for any K ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In [34, Conj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6], the authors conjecture the validity of the gluing theorem for non- collapsed RCD(K, N), with N the Hausdorff dimension of the metric-measure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' As introduced in [21], a non-collapsed RCD(K, N) space is an infinitesimally Hilbertian CD(K, N) space with m = H N, where H N denotes the N-dimensional Hausdorff mea- sure of (X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Since the weighted half-Grushin spaces are indeed collapsed, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='12 also shows that the non-collapsing assumption cannot be removed from [34, Conj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We wish to thank Michel Bonnefont for fruitful discussions and, in particular, for bringing some technical details in [23] that inspired the strategy of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2 to our attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 945655) and the ANR grant ‘RAGE’ (ANR-18-CE40-0012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The second-named author is member of the Istituto Nazionale di Alta Matematica (INdAM), Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA), and is par- tially supported by the INdAM–GNAMPA 2022 Project Analisi geometrica in strutture subriemanniane, codice CUP_E55F22000270001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Preliminaries In this section, we introduce some notation and recall some results about sub-Rieman- nian manifolds and curvature-dimension conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Sub-Riemannian structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' For L ∈ N, we let F = {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , XL} be a family of smooth vector fields globally defined on a smooth n-dimensional manifold M, n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The (generalized) sub-Riemannian distribution induced by the family F is defined by D = � x∈M Dx, Dx = span{X1|x, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , XL|x} ⊂ TxM, x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1) 8 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' RIZZI AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' STEFANI Note that we do not require the dimension of Dx to be constant as x ∈ M varies, that is, we may consider rank-varying distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' With a standard abuse of notation, we let Γ(D) = C∞-module generated by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Notice that, for any smooth vector field V , it holds V ∈ Γ(D) =⇒ Vx ∈ Dx for all x ∈ M, but the converse is false in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We let ∥V ∥x = min � |u| : u ∈ RL such that V = L � i=1 ui Xi|x, Xi ∈ F � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2) whenever V ∈ D and x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The norm ∥ · ∥x induced by the family F satisfies the parallelogram law and, consequently, it is induced by a scalar product gx : Dx × Dx → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' An admissible curve is a locally Lipschitz in charts path γ : [0, 1] → M such that there exists a control u ∈ L∞([0, 1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' RL) such that ˙γ(t) = L � i=1 ui(t)Xi|γ(t) for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The length of an admissible curve γ is defined via the norm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2) as length(γ) = � 1 0 ∥˙γ(t)∥γ(t) dt and the Carnot–Carathéodory (or sub-Riemannian) distance between x, y ∈ M is d(x, y) = inf{length(γ) : γ admissible with γ(0) = x, γ(1) = y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We assume that the family F satisfies the bracket-generating condition TxM = {X|x : X ∈ Lie(F)} for all x ∈ M, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3) where Lie(F) is the smallest Lie subalgebra of vector fields on M containing F, namely, Lie(F) = span � [Xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , [Xij−1, Xij]] : Xiℓ ∈ F, j ∈ N � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Under the assumption (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3), the Chow–Rashevskii Theorem implies that d is a well-defined finite distance on M inducing the same topology of the ambient manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Gradient, sub-Laplacian and Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The gradient of a function u ∈ C∞(M) is the unique vector field ∇u ∈ Γ(D) such that g(∇u, V ) = du(V ) for all V ∈ Γ(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4) One can check that ∇u can be globally represented as ∇u = L � i=1 Xiu Xi, with ∥∇u∥2 = L � i=1 (Xiu)2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5) even if the family F is not linearly independent, see Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2 for a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS 9 We equip the manifold M with a positive smooth measure m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The sub-Laplacian of a function u ∈ C∞(M) is the unique function ∆u ∈ C∞(M) such that � M g(∇u, ∇v) dm = − � M v ∆u dm (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6) for all v ∈ C∞ c (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' On can check that ∆u can be globally represented as ∆u = L � i=1 � X2 i u + Xiu divm(Xi) � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7) see Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2 for a proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7), divmV is the divergence of the vector field V computed with respect to m, that is, � M v divm(V ) dm = − � M g(∇v, V ) dm for all v ∈ C∞ c (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' For q ∈ [1, ∞), we say that u ∈ L1 loc(M, m) has q-integrable distributional Xi-derivative if there exists a function Xiu ∈ Lq(M, m) such that � M vXiu dm = � M uX∗ i v dm for all v ∈ C∞ c (M), where X∗ i v = −Xiv − v divm(Xi) denotes the adjoint action of Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We thus let HW1,q(M, m) = {u ∈ Lq(M, m) : Xiu ∈ Lq(M, m), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', L} be the usual horizontal W1,q Sobolev space induced by the the family F and the measure m on M, endowed with the natural norm ∥u∥HW1,q(M,m) = � ∥u∥q Lq(M,m) + ∥∇u∥q Lq(M,m) �1/q for all u ∈ HW1,q(M, m), where ∇u = L � i=1 Xiu Xi in accordance with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5) and ∥∇u∥q Lq(M,m) = � M ∥∇u∥q dm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Privileged coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Following [18,30], we introduce privileged coordinates, a fundamental tool in the description of the tangent cone of sub-Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Given a multi-index I ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', L}×i, i ∈ N, we let |I| = i be its length and we set XI = [XI1, [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', [XIi−1, XIi]]]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Accordingly, we define Di x = span{XI|x : |I| ≤ i} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='8) and ki(x) = dim Di x for all x ∈ M and i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, D0 x = {0} and D1 x = Dx as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1) for all x ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The spaces defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='8) naturally yield the filtration {0} = D0 x ⊂ D1 x ⊂ · · · ⊂ Ds(x) x = TxM for all x ∈ M, where s = s(x) ∈ N is the step of the sub-Riemannian structure at the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We say that x ∈ M is a regular point if the dimension of each space Di y remains constant as y ∈ M varies in an open neighborhood of x, otherwise x is a singular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 10 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' RIZZI AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' STEFANI Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1 (Adapted and privileged coordinates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let o ∈ M and let U ⊂ M be an open neighborhood of o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We say that the local coordinates given by a diffeomorphism z : U → Rn are adapted at o if they are centered at o, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' z(o) = 0, and ∂z1|0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', ∂zki|0 form a basis for Di o in these coordinates for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', s(o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We say that the adapted coordinate zi has weight wi = j if ∂zi|0 ∈ Dj o \\ Dj−1 o .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Furthermore, we say that the coor- dinates z are privileged at o if they are adapted at o and, in addition, zi(x) = O(d(x, o)wi) for all x ∈ U and i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Privileged coordinates exist in a neighborhood of any point, see [18, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Nilpotent approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' From now on, we fix a set of privileged coordinates z : U → Rn around a point o ∈ M in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Without loss of generality, we identify the coordinate domain U ⊂ M with Rn and the base point o ∈ M with the origin 0 ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Similarly, the vector fields in F defined on U are identified with vector fields on Rn, and the restriction of the sub-Riemannian distance d to U is identified with a distance function on Rn, which is induced by the family F, for which we keep the same notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' On (Rn, F), we define a family of dilations, for λ ≥ 0, by letting dilλ : Rn → Rn, dilλ(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , zn) = (λw1z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , λwnzn) for all z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , zn) ∈ Rn, where the wi’s are the weights given by Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We say that a differential operator P is homogeneous of degree −d ∈ Z if P(f ◦ dilλ) = λ−d(Pf) ◦ dilλ for all λ > 0 and f ∈ C∞(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='9) Note that the monomial zi is homogeneous of degree wi, while the vector field ∂zi is homogeneous of degree −wi, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' As a consequence, the differential operator zµ1 1 · · · · · zµn n ∂|ν| ∂zν1 1 · · · ∂zνn n , νi, µj ∈ N ∪ {0}, is homogeneous of degree �n i=1 wi(µi − νi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' For more details, see [18, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We can now introduce the new family � F = �� X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , � XL � by defining � Xi = lim ε→0 Xε i , Xε i = ε (dil1/ε)∗Xi, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='10) for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , L, where (dil1/ε)∗ stands for the usual push-forward via the differential of the dilation map dil1/ε, see [18, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The convergence in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='10) can be actually made more precise, in the sense that Xε i = � Xi + Rε i, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', L, where Rε i locally uniformly converges to zero as ε → 0, see [18, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The family � F is a set of complete vector fields on Rn, homogeneous of degree −1, with polynomial coefficients, and can be understood as the ‘principal part’ of F upon blow-up by dilations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Since F satisfies the bracket-generating condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3), also the new family � F is bracket-generating at all points of Rn, and thus induces a finite sub-Riemannian distance �d, see [18, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The resulting n-dimensional sub-Riemannian structure (Rn, � F ) is called nilpotent approximation of (Rn, F) at 0 ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS 11 The family � F = �� X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', � XL � generates a finite-dimensional stratified Lie algebra g = Lie( � F ) = g1 ⊕ · · · ⊕ gs of step s = s(0) ∈ N, where the grading is given by the degree of the vector fields, according to the definition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='9), that is, the layer gi corresponds to vector fields homogeneous of degree −i with respect to dilations, see [18, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, g1 = span �� X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , � XL � , so that g is generated by its first stratum, namely, gj+1 = [g1, gj], ∀j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='11) Finally, define the Lie subalgebra of vector fields vanishing at 0, h = �� X ∈ g : � X|0 = 0 � = h1 ⊕ · · · ⊕ hs, which inherits the grading from the one of g, hj+1 = [h1, hj], ∀j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='12) It is a fundamental fact [18, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='21] that the nilpotent approximation (Rn, � F ) is diffeo- morphic to the homogeneous sub-Riemannian space G/H, where G is the Carnot group G = exp g (explicitly realized as the subgroup of the flows of the vector fields of g acting on Rn from the right) and H = exp h is the Carnot subgroup induced by h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, if 0 ∈ Rn is a regular point, then H = {0}, and so the nilpotent approxi- mation (Rn, � F ) is diffeomorphic to the Carnot group G, see [18, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Recall that the smooth measure m on the original manifold M can be identified with a smooth measure on U ≃ Rn, for which we keep the same notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, m is absolutely continuous with respect to the n-dimensional Lebesgue measure L n on Rn, with m = ρ L n for some positive smooth function ρ: Rn → (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The corresponding blow-up measure on the nilpotent approximation is naturally given by �m = lim ε→0 mε = ρ(0) L n, mε = εQ (dil1/ε)#m, in the sense of weak∗ convergence of measures in Rn, where Q = n � i=1 i wi ∈ N is the so-called homogeneous dimension of (Rn, � F ) and (dil1/ε)# stands for the push- forward in the measure-theoretic sense via the dilation map dil1/ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Consequently, without loss of generality, we can assume that ρ(0) = 1, thus endowing (Rn, � F ) with the n- dimensional Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Notice that divL n � Xi = 0, for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , L, since each � Xi is homogeneous of degree −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Hence, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7), the sub-Laplacian of a function u ∈ C∞(Rn) can be globally represented as �∆u = L � i=1 � X 2 i u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='13) It is worth noticing that the metric space (Rn, �d ) induced by the nilpotent approxi- mation (Rn, � F ) actually coincides with the metric tangent cone at o ∈ M of the metric space (M, d) in the sense of Gromov [26], see [18, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='36] for the precise statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 12 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' RIZZI AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' STEFANI In fact, the sub-Riemmanian distance dε induced by the vector fields Xε i , i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , L, defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='10) is uniformly converging to the distance �d on compact sets as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' It is not difficult to check that the family {(Rn, dε, mε, 0)}ε>0 of pointed metric-measure spaces converge to the pointed metric-measure space (Rn, �d, L n, 0) as ε → 0 in the pointed measure Gromov–Hausdorff topology, see [13, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The curvature-dimension condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We end this section by recalling the defi- nition of curvature-dimension conditions of introduced in [36,48,49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' On a Polish (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', separable and complete) metric space (X, d), we let P(X) be the set of probability Borel measures on X and define the Wasserstein (extended) distance W2 W2 2(µ, ν) = inf �� X×X d2(x, y) dπ : π ∈ Plan(µ, ν) � ∈ [0, ∞], for µ, ν ∈ P(X), where Plan(µ, ν) = {π ∈ P(X × X) : (p1)#π = µ, (p2)#π = ν}, where pi : X ×X → X, i = 1, 2, are the projections on each component and T#µ ∈ P(Y ) denotes the push-forward measure given by any µ-measurable map T : X → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The function W2 is a distance on the Wasserstein space P2(X) = � µ ∈ P(X) : � X d2(x, x0) dµ(x) < ∞ for some, and thus any, x0 ∈ X � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Note that (P2(X), W2) is a Polish metric space which is geodesic as soon as (X, d) is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In addition, letting Geo(X) be the set of geodesics of (X, d), namely, curves γ : [0, 1] → X such that d(γs, γt) = |s−t| d(γ0, γ1), for all s, t ∈ [0, 1], any W2-geodesic µ: [0, 1] → P2(X) can be (possibly non-uniquely) represented as µt = (et)♯ν for some ν ∈ P(Geo(X)), where et: Geo(X) → X is the evaluation map at time t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We endow the metric space (X, d) with a non-negative Borel measure m such that m is finite on bounded sets and supp(m) = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We define the (relative) entropy functional Entm : P2(X) → [−∞, +∞] by letting Entm(µ) = � X ρ log ρ dm if µ = ρm and ρ log ρ ∈ L1(X, m), while we set Entm(µ) = +∞ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2 (CD(K, ∞) property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We say that a metric-measure space (X, d, m) satisfies the CD(K, ∞) property if, for any µ0, µ1 ∈ P2(X) with Entm(µi) < +∞, i = 0, 1, there exists a W2-geodesic [0, 1] ∋ s �→ µs ∈ P2(X) joining them such that Entm(µs) ≤ (1 − s) Entm(µ0) + s Entm(µ1) − K 2 s(1 − s) W2 2(µ0, µ1) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='14) for every s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The geodesic K-convexity of Entm in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='14) can be reinforced to additionally encode an upper bound on the dimension on the space, as recalled below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' For N ∈ (1, ∞), we let SN(µ, m) = − � X ρ−1/N dµ, µ = ρm + µ⊥, be the N-Rényi entropy of µ ∈ P2(X) with respect to m, where µ = ρm + µ⊥ denotes the Radon–Nikodym decomposition of µ with respect to m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS 13 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3 (CD(K, N) property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We say that a metric-measure space (X, d, m) satisfies the CD(K, N) property for some N ∈ [1, ∞) if, for any µ0, µ1 ∈ P2(X) with µi = ρim, i = 0, 1, there exists a W2-geodesic [0, 1] ∋ s �→ µs ∈ P2(X) joining them, with µs = (es)♯ν for some ν ∈ P(Geo(X)) such that SN′(µs, m) ≤ − � Geo(X) � τ (1−s) K,N′ (d(γ0, γ1))ρ−1/N′ 0 (γ0) + τ (s) K,N′(d(γ0, γ1))ρ−1/N′ 1 (γ1) � dν(γ) for every s ∈ [0, 1], N′ ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Here τ (s) K,N is the model distortion coefficient, see [49, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 137].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The CD(0, N) corresponds to the convexity of the N′-Rényi entropy SN′(µs, m) ≤ (1 − s)SN′(µ0, m) + sSN′(µ1, m), for every s ∈ [0, 1] and N′ ≥ N, with µ0, µ1 ∈ P2(X) as in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' For a CD(K, N) metric-measure space, K and N represent a lower bound on the Ricci tensor and an upper bond on the dimension, respectively, and we have CD(K, N) =⇒ CD(K, N′) for all N′ ≥ N, N, N′ ∈ [1, ∞], CD(K, N) =⇒ CD(K′, N) for all K′ ≤ K, K, K′ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, the CD(K, ∞) condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='14) is the weakest of all the curvature-dimension conditions for fixed K ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proofs We first deal with Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5, from which Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2 immediately follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We divide the proof in four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Step 1: passing to the nilpotent approximation via blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let (Rn, � F ) be the nilpotent approximation of (M, F) at some fixed point o ∈ M as explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let u ∈ C∞ c (M) and, without loss of generality, let us assume that supp u is contained in the domain of the privileged coordinates at o ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, we identify u with a C∞ c function on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We now apply (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1) to the dilated function uε = u ◦ dil1/ε ∈ C∞ c (Rn), for ε > 0, and evaluate this expression at the point dilε(x) ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Exploiting the expressions in Corol- lary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2, we get that L � i,j=1 Xε i u � Xε ijju − Xε jjiu � − (Xε iju)2 + Rε i,j u ≤ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1) where Xε i is as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='10) , Xijk = XiXjXk whenever i, j, k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , L}, and Rε i,j is a reminder locally uniformly vanishing as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Therefore, letting ε → 0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1), by the convergence in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='10) we get L � i,j=1 � Xiu �� Xijju − � Xjjiu � − �� Xiju �2 ≤ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2) which is (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1) with K = 0 for the nilpotent approximation (Rn, � F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 14 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' RIZZI AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' STEFANI Step 2: improvement via homogeneous structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We now show that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2) implies a stronger identity, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4) below, obtained from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2) by removing the squared term and replacing the inequality with an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Recall, in particular, the definition of weight of (privileged) coordinates in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We take u ∈ C∞(Rn) of the form u = α + γ, where α and γ are homogeneous polynomial of weighted degree 1 and at least 3, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Since XIα = 0 as soon as the multi-index satisfies |I| ≥ 2 (see [18, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='10]), we can take the terms with lowest homogeneous degree in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2) to get L � i,j=1 � Xiα �� Xijjγ − � Xjjiγ � = L � i=1 � Xiα �� Xi, �∆ � (γ) ≤ 0 for all such α and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In the second equality, we used the fact that the sub-Laplacian �∆ is a sum of squares as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Since α can be replaced with −α, we must have that L � i=1 � Xiα �� Xi, �∆ � (γ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3) Observing that � Xiα is homogeneous of degree 0, and thus a constant function, we can rewrite (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3) as � L � i=1 � Xiα � Xi, �∆ � (γ) = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4) which is the seeked improvement of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Step 3: construction of the space i ⊂ g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let Pn 1 be the vector space of homogeneous polynomials of weighted degree 1 on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Notice that Pn 1 = span{zi | i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , k1}, k1 = dim D|0, that is, Pn 1 is generated by the monomials given by the coordinates of lowest weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We now define a linear map φ: Pn 1 → g1 by letting φ[α] = � ∇α = L � i=1 � Xiα � Xi for all α ∈ Pn 1 (recall Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We claim that φ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Indeed, if φ[α] = 0 for some α ∈ Pn 1, then, by applying the operator φ[α] to the polynomial α, we get 0 = φ[α](α) = � L � i=1 � Xiα � Xi � (α) = L � i=1 (� Xiα)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Thus � Xiα = 0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Hence α must have weighted degree at least 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' However, since α is homogeneous of weighted degree 1, we conclude that α = 0, proving that ker φ = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We can thus define the subspace i = φ[Pn 1] ⊂ g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS 15 By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4), any � X ∈ i is such that [� X, �∆](γ) = 0 for any homogeneous polynomial γ of degree at least 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Exploiting the definitions given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4, we observe that a differential operator P, homogeneous of weighted degree −d ∈ Z, has the form P = � µ,ν aµ,νzµ ∂|ν| ∂zν , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5) where µ = (µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , µn), ν = (ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , νn), µi, νj ∈ N ∪ {0}, aµ,ν ∈ R, and the weighted degree of every addend in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5) is equal to −d, namely, �n i=1(µi − νi)wi = −d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Thus, since � X and �∆ are homogeneous differential operators of order −1 and −2, respectively, then [� X, �∆] has order −3, see [18, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' It follows that [� X, �∆] = 0 as differential operator acting on C∞(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We now show (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let us first observe that i ∩ h = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Indeed, if φ[α] ∈ h for some α ∈ Pn 1, that is, φ[α]|0 = 0, then � Xiα|0 = 0 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Since � Xiα is a constant function, this implies φ[α] = 0, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Therefore, since dim i = dim Pn 1 = k1, we must have g1 = i ⊕ h1 thanks to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' With the same notation of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4, if g1 = v ⊕ h1, then dim v = k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We claim that the dimension of v is preserved by evaluation at zero, that is, dim v|0 = dim v, where dim v|0 is the dimension of v|0 as a subspace of T0Rn, while dim v is the dimension of v as a subspace of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Indeed, we have the trivial inequality dim v|0 ≤ dim v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' On the other hand, if strict inequality holds, then v must contain non- zero vector fields vanishing at zero, contradicting the fact that v ∩ h = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Therefore, since dim g1|0 = k1 and dim h1|0 = 0, we get dim v = dim v|0 = k1 as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' □ Step 4: proof of the Killing property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We have so far proved the existence of i such that g1 = i ⊕ h1, and such that any element Y ∈ i commutes with the sub-Laplacian �∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We now show that all such Y is a Killing vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let Y ∈ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Since [Y, �∆] = 0, the induced flow φY s , for s ∈ R, commutes with �∆ when acting on smooth functions, that is, �∆(u ◦ φY s ) = ( �∆u) ◦ φY s (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6) for all u ∈ C∞(Rn) and s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Recall the sub-Riemannian Hamiltonian � H : T ∗Rn → R, � H(λ) = 1 2 L � i=1 ⟨λ, � Xi⟩2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7) for all λ ∈ T ∗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='13), � H is the principal symbol of �∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Thus, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6) it follows � H ◦ � φY s �∗ = � H, for all s ∈ R, where the star denotes the pull-back, and thus � φY s �∗ is a diffeomorphism on T ∗Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' This means that φY s is an isometry, as we now show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Indeed, for any given x ∈ Rn, the restriction � H|T ∗ x Rn is a quadratic form on T ∗ xRn, so (φY s )∗ must preserve its kernel, that is, (φY s )∗ ker � H|T ∗ φYs (x)Rn = ker � H|T ∗ x Rn (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='8) 16 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' RIZZI AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' STEFANI for all x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7), it holds ker � H|T ∗ x Rn = � D⊥ x , where ⊥ denotes the annihilator of a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' By duality, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='8) we obtain that (φY s )∗ � Dx = � DφYs (x) for all x ∈ Rn as required by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Finally, for λ ∈ T ∗ xM, let λ# ∈ Dx be uniquely defined by gx(λ#, V ) = ⟨λ, V ⟩x for all V ∈ Dx, and notice that the map λ �→ λ# is surjective on Dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Then it holds ∥λ#∥2 x = 2� H(λ), see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Thus, since (φY s )∗ preserves � H, the map (φY s )∗ preserves the sub-Riemannian norm, and thus g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' This means that φY s is an isometry, concluding the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We claim that gj = hj for all j ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='9) Note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='9) is enough to conclude the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5, since, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='9) combined with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='11) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='12), we immediately get that g = g1 ⊕ h2 ⊕ · · · ⊕ hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, we deduce that g|0 = g1|0, which in turn implies that g must be commu- tative, otherwise the bracket-generating condition would fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' To prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='9), we proceed by induction on j ≥ 2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof of the base case j = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We begin by proving the base case j = 2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' To this aim, let � X ∈ i and �Y ∈ g1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' By definition of Lie bracket, we can write � φ � X −s � ∗ �Y = s �� X, �Y � + o(s) as s → 0, where φ � X s , for s ∈ R, is the flow of � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Since g1|x = � D|x for all x ∈ Rn, and since � X is Killing (in particular (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2) holds for its flow), we have that [� X, �Y ]|x ∈ � D|x for all x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Since [� X, �Y ] ∈ g2 and so, in particular, [� X, �Y ] is homogeneous of degree −2, we have [� X, �Y ]|0 = � j : wj=2 aj ∂zj|0, for some constants aj ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' But we also must have that [� X, �Y ]|0 ∈ � D|0 and so, since � D|0 = span � ∂zj : wj = 1 � according to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1, [� X, �Y ]|0 = 0, that is, [� X, �Y ] ∈ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We thus have proved that [i, g1] ⊂ h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, since g1 = i ⊕ h1, we get [i, i] ⊂ h2 and [i, h1] ⊂ h2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='10) from which we readily deduce (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='9) for j = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof of the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let us assume that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='9) holds for some j ∈ N, j ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Since g1 = i ⊕ h1, by the induction hypothesis we can write gj+1 = [g1, gj] = [g1, hj] = [i, hj] + [h1, hj] = [i, hj] + hj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We thus just need to show that [i, hj] ⊂ hj+1 for all j ∈ N with j ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Note that we actually already proved the case j = 1 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Again arguing by induction (taking j = 1 as base case), by the Jacobi identity and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='10) we have [i, hj+1] = [i, [h1, hj]] = [h1, [hj, i]] + [hj, [i, h1]] ⊂ [h1, hj+1] + [hj, h2] = hj+2 FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS 17 as desired, concluding the proof of the induction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2 (Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5 in the case h = {0}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5 is much simpler if the nilpotent approximation (Rn, � F) is a Carnot group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', h = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Indeed, in this case, the base case j = 2 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='9) immediately implies that g2 = h2 = {0}, which in turn gives g = g1, so that g is commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In the following, we assume that the reader is familiar with the notions of upper gradient and of q-upper gradient, see [5] for the precise definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The next two lemmas are proved in [27] for sub-Riemannians structures on Rn equipped with the Lebesgue measure, and are immediately extended to the weighted case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let (M, d, m) be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' If u ∈ C(M) and 0 ≤ g ∈ L1 loc(M, m) be an upper gradient of u, then u ∈ HW1,1 loc(M, m) with ∥∇u∥ ≤ g m-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, if u ∈ Lip(M, d), then ∥∇u∥ ≤ Lip(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Without loss of generality we may assume that M = Ω ⊂ Rn is a bounded open set, the sub-Riemannian structure is induced by a family of smooth bracket-generating vector fields F = {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', XL} on Ω and m = θL n, where θ: Ω → [0, ∞) is smooth and satisfies 0 < infΩ θ ≤ supΩ θ < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Hence, L1(Ω, θL n) = L1(Ω, L n) as sets, with equivalent norms, so that 0 ≤ g ∈ L1 loc(Ω, L n) is an upper gradient of u ∈ C(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Hence, by [27, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7], we get that u ∈ HW1,1 loc(Ω, L n), with ∥∇u∥ ≤ g L n-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', and thus θL n-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' By definition of distributional derivative, we can write � Ω v Xiu dx = � Ω u [−Xiv + div(Xi)v] dx, ∀ v ∈ C1 c (Ω), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , L, where div denotes the Euclidean divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We apply the above formula with test function v = θw, for any w ∈ C1 c (Ω), getting � Ω w Xiu θ dx = � Ω u � −Xiw + div(Xi)w + Xiθ θ w � θ dx, ∀ w ∈ C1 c (Ω), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' , L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The function within square brackets is the adjoint X∗ i w with respect to the measure θL n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' It follows that HW1,q(Ω, θL n) = HW1,q(Ω, L n) as sets, with equivalent norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, u ∈ W1,1 D,loc(Ω, θL n) as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4 (Meyers–Serrin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let (M, d, m) be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='8 and let q ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Then HW1,q(M, m) ∩ C∞(M) is dense in HW1,q(M, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Up to a partition of unity and exhaustion argument, we can reduce to the case M = Ω ⊂ Rn is a bounded open set and m = θL n, where θ: Ω → [0, ∞) is as in the previous proof, so that HW1,q(Ω, L n) = HW1,q(Ω, θL n) as sets, with equivalent norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, we can assume that θ ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' This case is proved in [27, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let (M, d, m) be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='8 and let q ∈ [1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' If u ∈ HW1,q(M, m), then ∥∇u∥ is the minimal q-upper gradient of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let us first prove that ∥∇u∥ is a q-upper gradient of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Indeed, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4, we can find (uk)k∈N ⊂ HW1,q(M, m) ∩ C∞(M) such that uk → u in HW1,q(M, m) as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 18 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' RIZZI AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' STEFANI It is well-known that the sub-Riemannian norm of the gradient of a smooth function is an upper gradient, see [27, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Thus, for uk it holds |uk(γ(1)) − uk(γ(0))| ≤ � γ ∥∇uk∥ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Arguing as in [28, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 179], using Fuglede’s lemma (see [28, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5 and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 10]), we pass to the limit for k → ∞ in the previous equality, outside a q-exceptional family of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' This proves that any Borel representative of ∥∇u∥ is a q-upper gradient of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We now prove that ∥∇u∥ is indeed minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let 0 ≤ g ∈ Lq(M, m) be any q-upper gradient of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Arguing as in [28, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 194], we can find a sequence (gk)k∈N ⊂ Lq(M, m) of upper gradients of u such that gk ≥ g for all k ∈ N and gk → g both pointwise m-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' on M and in Lq(M, m) as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3, we thus must have that ∥∇u∥ ≤ gk m-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' on M for all k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Hence, passing to the limit, we conclude that ∥∇u∥ ≤ g m-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' on M for any q-upper gradient g, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' □ We are now ready to deal with the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Recall that, here, q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We begin by claiming that W1,q(M, d, m) ⊂ HW1,q(M, m) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='11) isometrically, with ∥∇u∥ = |Du|w,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Indeed, let u ∈ W1,q(M, d, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' By a well-known approximation argument, combining [5, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3 and Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4], we find (uk)k∈N ⊂ Lip(M, d) ∩ W1,q(M, d, m) such that uk → u and |Duk|w,q → |Du|w,q in Lq(M, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='12) Since uk ∈ Lip(M, d), by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3 we know that uk ∈ HW1,q(M, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Hence, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5, |Duk|w,q = ∥∇uk∥, and we immediately get that sup k∈N � M ∥∇uk∥q dm < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Therefore, up to passing to a subsequence, (Xiuk)k∈N is weakly convergent in Lq(M, m), say Xiuk ⇀ αi ∈ Lq(M, m), for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We thus get that u ∈ HW1,q(M, m) with Xiu = αi and thus ∇u = �L i=1 αiXi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' By stability of q-upper gradients, [5, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3 and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4], ∥∇u∥ is a q-upper gradient of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' By semi-continuity of the norm, we obtain � M ∥∇u∥q dm ≤ lim inf k→∞ � M ∥∇uk∥q dm = � M |Du|q w,q dm, where we used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' By definition of minimal q-upper gradient we thus get that ∥∇u∥ = |Du|w,q m-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', and the claimed inclusion in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='11) immediately follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We now observe that it also holds HW1,q(M, m) ∩ C∞(M) ⊂ W1,q(M, d, m), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='13) with ∥∇u∥ = |Du|w,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We just need to notice that, if u ∈ C∞(M), then ∥∇u∥ is an upper gradient of u, see [27, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Therefore, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3, ∥∇u∥ must coincide with the minimal q-upper gradient of u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', ∥∇u∥ = |Du|w m-a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='13) readily follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In view of the isometric inclusions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='13), and of the density provided by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4, this concludes the proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' □ FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS 19 Proof of (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let us assume that (M, d, m) satisfies the CD(K, ∞) property for some K ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' By the previous point (i), we know that (M, d, m) satisfies the RCD(K, ∞) property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Consequently, since clearly C∞ c (M) ⊂ W1,2(M, d, m) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='13), [6, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3] (even if the measure m is σ-finite, see [4, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 7] for a discussion) implies that 1 2 � M ∆v ∥∇u∥2 dm − � M v g(∇u, ∇∆u) dm ≥ K � M v ∥∇u∥2 dm for all u, v ∈ C∞ c (M) with v ≥ 0 on M, from which we readily deduce (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The above proofs work for more general measures m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Namely, we can assume that, locally on any bounded coordinate neighborhood Ω ⊂ Rn, m = θL n with θ ∈ W1,1(Ω, L n) ∩ L∞(Ω, L n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In this case, the positivity of m corresponds to the requirement that θ is locally essentially bounded from below away from zero, in charts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We prove the two points in the statement separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof of (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The case p = 0 has been already considered by Juillet in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' For p > 0, we can argue as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let A0 = [−ℓ −1, −ℓ]×[0, 1] and A1 = [ℓ, ℓ + 1]×[0, 1] for ℓ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We will shortly prove that the midpoint set A1/2 = � q ∈ R2 : ∃ q0 ∈ A0, ∃ q1 ∈ A1 with d(q, q0) = d(q, q1) = 1 2 d(q0, q1) � satisfies A1/2 ⊂ [−1 − εℓ, 1 + εℓ] × [0, 1] (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='14) for some εℓ > 0, with εℓ ↓ 0 as ℓ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Since mp(A0) = mp(A1) ∼ ℓp as ℓ → ∞, we get � mp(A0) mp(A1) > mp(A1/2) for large ℓ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' This contradicts the logarithmic Brunn–Minkowski BM(0, ∞) inequality, which is a consequence of the CD(0, ∞) condition, see [50, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' To prove (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='14), let qi ∈ Ai, qi = (xi, yi), and let γ(t) = (x(t), y(t)), t ∈ [0, 1], be a geodesic such that γ(i) = qi, with i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We first note that min{y0, y1} ≤ y(t) ≤ max{y0, y1} for all t ∈ [0, 1], (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='15) since any curve that violates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='15) can be replaced by a strictly shorter one satisfy- ing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, we get that A1/2 ⊂ R × [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let us now observe that |xa − xb| ≤ d(a, b) ≤ |xa − xb| + |ya − yb| max{|xa|, |xb|} for all a = (xa, ya) and b = (xb, yb) with xa, xb ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Therefore, if q = (x, y) ∈ A1/2, then |x − x0| ≤ d(q, q0) = 1 2 d(q0, q1) ≤ ℓ + 1 + O(1/ℓ) and, similarly, |x − x1| ≤ ℓ + 1 + O(1/ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Since x0 ∈ [−ℓ − 1, −ℓ] and x1 ∈ [ℓ, ℓ + 1], we deduce that |x| ≤ 1 + O(1/ℓ), concluding the proof of the claimed (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' □ 20 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' RIZZI AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' STEFANI Proof of (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Out of the negligible set {x = 0}, the metric g on Gp given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5) is locally Riemannian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Recalling (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7), the BE(K, ∞) inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1) is implied by the lower bound Ric∞,V ≥ K via Bochner’s formula, where Ric∞,V is the ∞-Bakry–Émery Ricci tensor of (R2, g, e−V volg), see [50, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 14, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='36) – (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='51)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7 below, we have Ric∞,V ≥ 0 for all p ≥ 1, concluding the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let p ∈ R and N > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The N-Bakry–Émery Ricci tensor of the Grushin metric (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5), with weighted measure mp = |x|p dx dy, for all x ̸= 0 is RicN,V = p − 1 x2 g −(p + 1)2 N − 2 dx ⊗ dx x2 , with the convention that 1/∞ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The N-Bakry–Émery Ricci tensor of a n-dimensional weighted Riemannian struc- ture (g, e−V volg), for N > n, is given by RicN,V = Ricg + HessgV − dV ⊗ dV N − n , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='16) see [50, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='36)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In terms of the frame (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4), the Levi-Civita connection is given by ∇XX = ∇XY = 0, ∇Y X = −1 xY, ∇Y Y = 1 xX, whenever x ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Recalling that, from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7), V (x) = −(p + 1) log |x|, for x ̸= 0, we obtain Ricg = − 2 x2 g, HessgV = (p + 1) x2 g, dV = −p + 1 x dx, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='17) whenever x ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The conclusion thus follows by inserting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='17) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The statement is a consequence of the geodesic convexity of G+ p and the computation of the N-Bakry–Émery curvature in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Since the proof uses quite standard arguments, we simply sketch its main steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The interior of G+ p , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=', the open half-plane, can be regarded as a (non-complete) weighted Riemannian manifold with metric g as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5) and weighted volume as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let µ0, µ1 ∈ P2(G+ p ), µ0, µ1 ≪ mp, with bounded support contained in the Riemannian region {x > ε}, for some ε ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Let (µs)s∈[0,1] be a W2-geodesic joining µ0 and µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' By a well-known representation theorem (see [50, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='22]), there exists ν ∈ P(Geo(G+ p )), supported on the set Γ = (e0 × e1)−1(supp µ0 × supp µ1), such that µs = (es)♯ν for all s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Since the set {x ≥ ε} is a geodesically convex subset of the full Grushin plane Gp (by the same argument of [46, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 5]), any γ ∈ Γ is contained for all times in the region {x > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Therefore, Γ is a set of Riemannian geodesics contained in the weighted Riemannian struc- ture ({x > 0}, g, e−V volg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7, we have RicN,V ≥ 0 for all N ≥ Np, where Np is as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' At this point, a standard argument shows that the Rényi entropy is convex along Wasserstein geodesics joining µ0 with µ1, see the proof of [49, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The extension to µ0, µ1 ∈ P2(G+ p ), with µ0, µ1 ≪ mp and compact support possibly touching the singular region {x = 0}, is achieved via a standard approximation argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' More precisely, one reduces to the previous case and exploits the stability of optimal transport [50, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='9] and the lower semi-continuity of the Rényi entropy [50, Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS 21 Finally, the extension to general µ0, µ1 ∈ P2(G+ p ) follows the routine argument outlined in [9, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='12], which works when µs = (es)♯ν, s ∈ [0, 1], and ν is concentrated on a set of non-branching geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' This proves the ‘if’ part of the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The ‘only if’ part is also standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The CD(0, N) condition for N > 2 implies that, on the Riemannian region {x > 0}, RicN,V ≥ 0, but this is false for N < Np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The fact that G+ p is infinitesimally Hilbertian follows from Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='9, by noting that mp is positive and smooth out of the closed set {x = 0}, which has zero measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' An alternative proof follows from the observation that G+ p is a Ricci limit, see [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' □ Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Gradient and Laplacian representations formulas For the reader’s convenience, in this appendix we provide a short proof of the repre- sentation formulas (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7), in the rank-varying case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' For λ ∈ T ∗M, let λ# ∈ D be uniquely defined by g(λ#, V ) = ⟨λ, V ⟩ for all V ∈ D, where ⟨·, ·⟩ denotes the action of covectors on vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Then ∥λ#∥2 = L � i=1 � λ#, Xi �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1) As a consequence, if λ, µ ∈ T ∗M, then g(λ#, µ#) = L � i=1 ⟨λ, Xi⟩⟨µ, Xi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Given u ∈ RL, we set Xu = �L i=1 uiXi and define u∗ ∈ argmin � v �→ |v| : v ∈ RL, Xv = Xu � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In other words, for Xu ∈ D, u∗ is the element of minimal Euclidean norm such that Xu∗ = Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Note that, by definition, it holds ∥Xu∥ = |u∗|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We thus have ∥λ#∥ = sup � g(λ#, X) : ∥X∥ = 1, X ∈ D � = sup � g(λ#, Xu) : |u∗| = 1, u ∈ RL� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We now claim that sup � g(λ#, Xu) : |u∗| = 1, u ∈ RL� = sup � g(λ#, Xu) : |u| = 1, u ∈ RL� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3) Indeed, the inequality ≤ in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3) is obtained by observing that Xu = Xu∗ for any u ∈ RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' To prove the inequality ≥ in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3), we observe that, if u ∈ RL is such that |u| = 1 and 0 < |u∗| < 1, then v = u/|u∗| satisfies |v∗| = 1 and gives g(λ#, Xv) > g(λ#, Xv) |u∗| = g(λ#, Xu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4) Furthermore, if |u| = 1 and u∗ = 0, then Xu = 0 so also in this case we find v ∈ Rn with v∗ = 1 such that (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' This ends the proof of the claimed (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Hence, since g(λ#, Xu) = L � i=1 g(λ#, Xi) ui, 22 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' RIZZI AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' STEFANI we easily conclude that ∥λ#∥ = sup � g(λ#, Xu) : |u| = 1, u ∈ RL� = � � � � L � i=1 g(λ#, Xi)2, proving (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Equality (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2) then follows by polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' □ Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' The following formulas hold: ∇u = L � i=1 Xiu Xi, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5) ∆u = L � i=1 � X2 i u + Xiu divm(Xi) � , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6) g(∇u, ∇v) = L � i=1 Xiu Xiv, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7) for all u, v ∈ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' In particular, ∥∇u∥ = �L i=1(Xiu)2 for all u ∈ C∞(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' We prove each formula separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Recalling the definition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='4), we can pick λ = du in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2) to get � du, µ#� = g(∇u, µ#) = L � i=1 ⟨du, Xi⟩⟨µ, Xi⟩ = L � i=1 Xiu ⟨µ, Xi⟩ = g � µ#, L � i=1 Xiu Xi � whenever µ ∈ T ∗ xM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Since the map #: T ∗ xM → Dx is surjective, we immediately get (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Recall that divm(fX) = Xf + f divm(X) for any f ∈ C∞(M) and X ∈ Γ(TM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Hence, from the definition in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6), we can compute ∆u = divm(∇u) = L � i=1 divm(Xiu Xi) = L � i=1 � X2 i u Xi + Xiu divm(Xi) � , which is the desired (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Proof of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Choosing λ = du and µ = dv in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='2), we can compute g(∇u, ∇v) = L � i=1 ⟨du, Xi⟩ ⟨dv, Xi⟩ = L � i=1 Xiu Xiv and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' □ FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS 23 References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Agrachev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Barilari, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Rizzi, Curvature: a variational approach, Mem.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' [50] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Villani, Optimal transport, Grundlehren der mathematischen Wissenschaften [Fundamental Prin- ciples of Mathematical Sciences], vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' 338, Springer-Verlag, Berlin, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Old and new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' FAILURE OF CD CONDITIONS ON SUB-RIEMANNIAN MANIFOLDS 25 (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Rizzi) Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265, 34136 Trieste (TS), Italy Email address: lrizzi@sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='it (G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content=' Stefani) Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265, 34136 Trieste (TS), Italy Email address: gstefani@sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='it or giorgio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='stefani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='math@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tAyT4oBgHgl3EQf1flH/content/2301.00735v1.pdf'} diff --git a/29AyT4oBgHgl3EQfPvbO/content/tmp_files/2301.00032v1.pdf.txt b/29AyT4oBgHgl3EQfPvbO/content/tmp_files/2301.00032v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..bb529cb5002e173765936cf22de00f994c76c620 --- /dev/null +++ b/29AyT4oBgHgl3EQfPvbO/content/tmp_files/2301.00032v1.pdf.txt @@ -0,0 +1,1717 @@ +Bayesian Learning for Dynamic Inference +Aolin Xu +Peng Guan +Abstract +The traditional statistical inference is static, in the sense that the estimate of the quantity +of interest does not affect the future evolution of the quantity. In some sequential estimation +problems however, the future values of the quantity to be estimated depend on the estimate of +its current value. This type of estimation problems has been formulated as the dynamic inference +problem. In this work, we formulate the Bayesian learning problem for dynamic inference, +where the unknown quantity-generation model is assumed to be randomly drawn according to +a random model parameter. We derive the optimal Bayesian learning rules, both offline and +online, to minimize the inference loss. Moreover, learning for dynamic inference can serve as a +meta problem, such that all familiar machine learning problems, including supervised learning, +imitation learning and reinforcement learning, can be cast as its special cases or variants. Gaining +a good understanding of this unifying meta problem thus sheds light on a broad spectrum of +machine learning problems as well. +1 +Introduction +1.1 +Dynamic inference +Traditional statistical estimation, or statistical inference in general is static, in the sense that the +estimate of the quantity of interest does not affect the future evolution of the quantity. In some +sequential estimation problems however, we do encounter the situation where the future value of the +quantity to be estimated depends on the estimate of its current value. Examples include 1) stock +price prediction by big investors, where the prediction of the tomorrow’s price of a stock affects +tomorrow’s investment decision, which further changes the stock’s supply-demand status and hence +its price the day after tomorrow; 2) interactive product recommendation, where the estimate of +a user’s preference based on the user’s activity leads to certain product recommendations to the +user, which would in turn shape the user’s future activity and preference; 3) behavior prediction in +multi-agent systems, e.g. vehicles on the road, where the estimate of an adjacent vehicle’s intention +based on its current driving situation leads to a certain action of the ego vehicle, which can change +the future driving situation and intention of the adjacent vehicle. We may call such problems as +dynamic inference, which is formulated and studied in depth in [1]. It is shown that the problem of +dynamic inference can be converted to an Markov decision-making process (MDP), and the optimal +estimation strategy can be derived through dynamic programming. We give a brief overview of the +problem of dynamic inference in Section 2. +1.2 +Learning for dynamic inference +There are two major ingredients in dynamic inference: the probability transition kernels of the +quantity of interest given each observation, and the probability transition kernels of the next +1 +arXiv:2301.00032v1 [cs.LG] 30 Dec 2022 + +observation given the current observation and the estimate of the current quantity of interest. We +may call them the quantity-generation model and the observation-transition model, respectively. +Solving the dynamic inference problem requires the knowledge of the two models. However, in most +of the practically interesting situations, we do not have such knowledge. Instead, we either have a +training dataset from which we can learn these models or we can learn them on-the-fly during the +inference. +In this work, we set up the learning problem in a Bayesian framework, and derive the optimal +learning rules, both offline (Section 3) and online (Section 4), for dynamic inference under this +framework. Specifically, we assume the unknown models are elements in some parametric families +of probability transition kernels, and the unknown model parameters are randomly drawn according +to some prior distributions. The goal is then to find an optimal Bayesian learning rule, which can +return an estimation strategy that minimizes the inference loss. The approach we take toward this +goal is converting the learning problem to an MDP with an augmented state, which consists of +the current observation and a belief vector of the unknown parameters, and solving the MDP by +dynamic programming over the augmented state space. The solution, though optimal, may still be +computationally challenging unless the belief vector can be compactly represented. Nevertheless, it +already has a greatly reduced search space compared to the original learning problem, and provides +a theoretical basis for the design of more computationally efficient approximate solutions. +Perhaps equally importantly, the problem of learning for dynamic inference can serve as a meta +problem, such that almost all familiar learning problems can be cast as its special cases or variants. +Examples include supervised learning, imitation learning, and reinforcement learning, including +bandit and contextual bandit problems. For instance, the Bayesian offline learning for dynamic +inference can be viewed as an extension of the behavior cloning method in imitation learning [2–4], +in that it not only learns the demonstrator’s action-generation model, but simultaneously learns a +policy based on the learned model to minimize the overall imitation error. As another instance, the +quantity to be estimated in dynamic inference may be viewed as a latent variable of the loss function, +so that the Bayesian online learning for dynamic inference can be viewed as Bayesian reinforcement +learning [5–8], where an optimal policy is learned by estimating the unknown loss function. Learning +for dynamic inference thus provides us with a unifying formulation of different learning problems. +Having a good understanding of this problem is helpful for gaining better understandings of the +other learning problems as well. +1.3 +Relation to existing works +The problem of dynamic inference and learning for dynamic inference appear to be new, but it +can be viewed from different angles, and is related to a variety of existing problems. The most +intimately related work is the original formulations of imitation learning [9]. The online learning for +dynamic inference is closely related to ans subsumes Bayesian reinforcement learning. Some recent +study on Bayesian reinforcement learning and interactive decision making include [10,11]. +A problem formulation with a similar spirit in a minimax framework appear recently in [12]. In +that work, an adversarial online learning problem where the action in each round affects the future +observed data is set up. It may be viewed as adversarial online learning for dynamic minimax +inference, from our standpoint. The advantage of the Bayesian formulation is that all the variables +under consideration, including the unknown model parameters, are generated from some fixed joint +distribution, thus the optimality of learning can be defined and the optimal learning rule can be +derived. On the contrary, with the adversarial formulation, only certain definitions of regret can be +2 + +studied. +The overall optimality proof technique we adopt is similar to those used in solving partially +observed MDP (POMDP) and Bayesian reinforcement learning over the augmented belief space +[13,14]. Several proofs are adapted from the rigorous exposition of the optimality of the belief-state +MDP reformulation of the POMDP [15]. +As mentioned in the previous subsection, Bayesian learning for dynamic inference can be viewed +as a unifying formulation for Bayesian imitation learning and Bayesian reinforcement learning. +These problems are surveyed in [16–18] for relevant imitation learning, and in [8,19–22] for relevant +reinforcement learning. +2 +Overview of dynamic inference +2.1 +Problem formulation +The problem of an n-round dynamic inference is to estimate n unknown quantities of interest +Y n sequentially based on observations Xn, where in the ith round of estimation, Xi depends on +the observation Xi−1 and the estimate �Yi−1 of Yi−1 in the previous round, while the quantity of +interest Yi only depends on Xi, and the estimate �Yi of Yi can depend on everything available so +far, namely (Xi, �Y i−1), through an estimator ψi as �Yi = ψi(Xi, �Y i−1). The sequence of estimators +ψn = (ψ1, . . . , ψn) constitute an estimation strategy. We assume to know the distribution PX1 of the +initial observation, and the probability transition kernels (KXi|Xi−1,�Yi−1)n +i=2 and (KYi|Xi)n +i=1. These +distributions and ψn define a joint distribution of (Xn, Y n, �Y n), all the variables under consideration. +The Bayesian network of the random variables in dynamic inference with a Markov estimation +strategy, meaning that each estimator has the form ψi : X → �Y, is illustrated in Fig. 1. +Figure 1: Bayesian network of the random variables under consideration with n = 4. Here we +assume the estimates are made with Markov estimators, such that �Yi = ψi(Xi). +The goal of dynamic inference can then be formally stated as finding an estimation strategy to +minimize the accumulated expected loss over the n-rounds: +arg min +ψn +E +� +n +� +i=1 +ℓ(Xi, Yi, �Yi) +� +, +�Yi = ψi(Xi, �Y i−1) +(1) +where ℓ : X×Y× �Y → R is a loss function that evaluates the estimate made in each round. Compared +with the traditional statistical inference under the Bayesian formulation, where the goal is to find +an estimator ψ of a random quantity Y based on a jointly distributed observation X to minimize +E[ℓ(Y, ψ(X))], we summarize the two distinctive features of dynamic inference in (1): +3 + +Yi +Y2 +Y3 +Y4 +1 +X1 +★ X2 ++ X3 + 250 then +18 +Break; +19 +end +20 +end +/* record statistics of class pairs +*/ +21 +𝑇 = 𝜒2-Test-Statistic(𝑽); +22 +𝑻𝑐1𝑐2 ← 𝑻𝑐1𝑐2 +𝑇/𝐵; +23 +𝑻𝑐2𝑐1 ← 𝑻𝑐2𝑐1 +𝑇/𝐵; +24 +end +25 +end +26 end +27 Compute 𝑝-value table 𝑭𝑐×𝑐 with average statistics in 𝑻; +28 Return 𝑭; +Algorithm 1: Network Effect Analysis (NEA) +in most real-world node classification tasks only a few node la- +bels are observed. We propose NEA to address such limitations. +Before introducing NEA, we provide two propositions: +PROPOSITION 1. Given a graph and a class 𝑐𝑖, if the nodes +with class 𝑐𝑖 tend to connect uniformly to the nodes with all +classes 1, ...,𝑐 equally, then class 𝑐𝑖 has no NE. +PROPOSITION 2. If all classes 𝑐𝑖 = 1, ...,𝑐 in a graph have no +NE, then this graph has no NE. +We separate heterophily graphs from those with no NE by the +propositions. In heterophily graphs, the nodes of a specific class +are likely to be connected to the nodes of other classes, such as +in bipartite graphs that connect different classes of nodes. In this +case, knowing the label of a node gives meaningful information +about the labels about its neighbors. On the other hand, if a graph +has no NE, every node has equal probabilities for more than one +class even after we consider the structural information from its +neighbors, which is useless to infer its true label. +To analyze whether a specific class 𝑐𝑖 has NE or not, we use +𝜒2 test to identify whether there exists a statistically significant +contingency between the classes. Given two classes 𝑐1 and 𝑐2, the + +Under Submission, , +Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos +1 +2 +Class ID +1 +2 +Class ID +Edge Counting +105 +106 +2 × 105 +3 × 105 +4 × 105 +6 × 105 +1 +2 +Class ID +1 +2 +Class ID +p-value Table +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +(a) “Genius”: No NE +1 +2 +Class ID +1 +2 +Class ID +Edge Counting +6.6 × 105 +6.7 × 105 +6.8 × 105 +6.9 × 105 +7 × 105 +7.1 × 105 +7.2 × 105 +7.3 × 105 +7.4 × 105 +1 +2 +Class ID +1 +2 +Class ID +p-value Table +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +(b) “Penn94”: No NE +1 +2 +Class ID +1 +2 +Class ID +Edge Counting +3 × 106 +3.2 × 106 +3.4 × 106 +3.6 × 106 +3.8 × 106 +4 × 106 +1 +2 +Class ID +1 +2 +Class ID +p-value Table +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +(c) “Twitch”: No NE +1 +2 +3 +4 +5 +Class ID +1 +2 +3 +4 +5 +Class ID +Edge Counting +105 +3 × 104 +4 × 104 +6 × 104 +2 × 105 +1 +2 +3 +4 +5 +Class ID +1 +2 +3 +4 +5 +Class ID +p-value Table +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +(d) “arXiv-Year”: X-ophily with Weak NE +1 +2 +3 +4 +5 +Class ID +1 +2 +3 +4 +5 +Class ID +Edge Counting +105 +2 × 105 +3 × 105 +4 × 105 +6 × 105 +1 +2 +3 +4 +5 +Class ID +1 +2 +3 +4 +5 +Class ID +p-value Table +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +(e) “Patent-Year”: Heterophily with Weak NE +1 +2 +Class ID +1 +2 +Class ID +Edge Counting +107 +8 × 106 +9 × 106 +1.1 × 107 +1.2 × 107 +1.3 × 107 +1 +2 +Class ID +1 +2 +Class ID +p-value Table +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +(f) “Pokec-Gender”: Heterophily with Strong NE +Figure 2: NEA discovers that real-world heterophily graphs do not necessarily have network-effect (NE). For each dataset, +we report the edge counting on the left, and the 𝑝-value table output from NEA on the right. We have a case of X-ophily, e.g. +in “arXiv-Year”, class 1 is homophily, and the rest are heterophily. +input to the test is 2 × 2 contingency table with counts of edges +where nodes of each edge ∈ {𝑐1,𝑐2}. +NULL HYPOTHESIS 1. Edges are equally likely to exhibit +homophily and heterophilly. +Algorithm 1 presents the procedure for the proposed NEA. +A practical challenge is that if the numbers in the table are too +large, 𝑝-value becomes extremely small and meaningless [24]. +However, sampling for only a single round can be unstable and +output very different results. To address this, we combine 𝑝- +values from different random sampling by Universal Inference +[38]. We firstly sample edges to add to the contingency table +until the frequency is above a specified threshold, and compute +the 𝜒2 test statistic for each class pair. Next, following Universal +Inference, we repeat the procedure for random samples of edges +for 𝐵 rounds and average the statistics. At last, we use the average +statistics to compute the 𝑝-value table 𝑭𝑐×𝑐 of 𝜒2 tests. +It is worth noting that, NEA is robust to the noisy edges, thanks +to the random sampling. It also works well given either a few or +many node labels. Given only a few observations, 𝜒2 test works +well enough when the frequency in the contingency table are only +at least 5; given many observations, the sampling and combining +trick ensures the correctness of 𝑝-value. +We give observations based on the result of NEA: +OBSERVATION 1. If a class accepts all the null hypotheses in +Algorithm 1, then this class has no NE. +We then extend Observation 1 to an extreme case: +OBSERVATION 2. If all classes in a graph obey Observation 1, +the node classification problem is unsolvable under our setting. +3.2 +Discoveries +For each dataset, we equally sample 5% of node labels and com- +pute the 𝑝-value table by Algorithm 1. This is because a) only +a few labels are observed in most node classification tasks, and +thus it is natural to make the same assumption in this analysis, +and b) our NEA can correctly analyze NE even from partial ob- +servations. We set 𝐵 = 1000 to output stable results. Based on +Observation 2, here is our surprising discovery: +DISCOVERY 1 (NO NE). “Genius”, “Penn94”, and “Twitch” +have no NE, exhibiting neither homophily nor heterophily. +“Genius” [22], “Penn94” [34], and “Twitch” [31] have been +widely used in previous works [21, 23, 25, 27, 36, 40]. In “Ge- +nius” (Figure 2a), we see that both classes 1 and 2 tend to connect +to class 1. This makes the class 2 indistinguishable by the graph +structure. NEA thus accepts the null hypothesis and identifies +that there exists no statistically significant difference. This means +that the edges have the same probabilities to be homophily and +heterophily. We can see a similar phenomenon in “Penn94” (Fig- +ure 2b). “Twitch” (Figure 2c) is not considered as a homophily +graph because the effect is too weak, where the scales on the +color bar are very close. However, it is not a heterophily graph +as well, where NEA correctly identifies that every class tends to +connect to both classes near-uniformly. +We further analyzed three more datasets: +DISCOVERY 2 (WEAK AND STRONG NE). “Arxiv” and +“Patent-Year” exhibit weak NE; and “Pokec-Gender” exhibits +strong NE. +The “arXiv-Year” and “Patent-Year” datasets (Figure 2d and 2e) +have weak NE, where one of the classes accepts more than one +null hypothesis. “Pokec-Gender” (Figure 2f) shows strong NE, +where the estimated 𝑝-value is 0.008. These three datasets will +later be used in our experiments. +4 +PROPOSED METHOD PART II – +ULTRAPROP +We propose ULTRAPROP, our approach for accurate node classifi- +cation. Algorithm 2 shows the algorithm of ULTRAPROP. In line +1, given an adjacency matrix 𝑨 and rank 𝑑, we make “Emphasis” +Matrix 𝑨∗ (in Section 4.1) to handle the neighbor-differentiation +(ND). To handle network-effect (NE), we estimate the compati- +bility matrix ˆ𝑯∗ from 𝑨∗ in line 2 (in Section 4.2). In line 3 to 7, +we initialize and propagate the beliefs ˆ𝑩 iteratively through 𝑨∗ +until they converge. In each iteration, we aggregate the beliefs of +neighbors in ˆ𝑩, weighted by the values in 𝑨∗. This aims to draw +attention to the neighbors that are more structurally important. + +ULTRAPROP: Principled and Explainable Propagation on Large Graphs +Under Submission, , +Data: Adjacency matrix 𝑨, initial belief ˆ𝑬, priors P, and +decomposition rank 𝑑 +Result: Final belief 𝑩 +1 𝑨∗ ← “Emphasis”-Matrix(𝑨,𝑑); +2 ˆ𝑯∗ ← Compatibility-Matrix-Estimation(𝑨∗, ˆ𝑬, P); +/* propagation +*/ +3 ˆ𝑩(0) ← 𝑶𝑛×𝑐,𝑡 ← 0; +4 while inferences changed and +� | ˆ𝑩(𝑡+1)− ˆ𝑩(𝑡) | +𝑛𝑐 +> +1 +lg𝑛𝑐 do +5 +ˆ𝑩(𝑡+1) ← ˆ𝑬 + 𝑓 𝑨∗ ˆ𝑩(𝑡) ˆ𝑯∗; +6 +𝑡 ← 𝑡 + 1; +7 end +8 Return 𝑩 ← ˆ𝑩(𝑡) + 1 +𝑐 ; +Algorithm 2: ULTRAPROP +The interrelations between classes is handled by multiplying with +ˆ𝑯∗. We further include an early stopping criterion in line 4 for +more efficient propagation. +4.1 +“Emphasis” Matrix +To incorporate the idea of ND, where neighbors have different +importances, we propose to replace the unweighted adjacency +matrix 𝑨 with a weighted one. The weight of edge (𝑖, 𝑗) reflects +the influence of node 𝑖 for 𝑗. We present an efficient solution to +weigh 𝑨 without using any node labels. It firstly embeds nodes +into structure-aware representations via random walks, and then +measures their similarities via distances in the embedding space. +Structure-Aware Node Representation. We represent nodes +in 𝑑-dimensional vector space efficiently using Singular Value +Decomposition (SVD) on the high-order proximity matrix of the +graph and capture information from pairwise connections. To +fast approximate the higher-order proximity matrix, we utilize +random walks described in Algorithm 3 from line 1 to 8. Given a +proximity matrix 𝑾 ′, 𝑾 ′ +𝑖𝑗 records the number of times we visit +node 𝑗 if we start a random walk from node 𝑖. Each neighbor has +the same probability of being visited in the unweighted graphs, +where only those structurally important neighbors are visited +more frequently. +To theoretically justify why it works, we prove that the neigh- +bor distribution for each node converges after a number of trials: +LEMMA 1 (CONVERGENCE OF REGULAR RANDOM WALKS). +With probability 1−𝛿, the error 𝜖 between the approximated distri- +bution and the true one for a node walking to its 1-hop neighbor +by a regular random walk of length 𝐿 with 𝑀 trials is less than +𝜖 ≤ ⌈(𝐿 − 1)/2⌉ +𝐿 +√︂ +log (2/𝛿) +2𝐿𝑀 +(2) +PROOF. Omitted for brevity. Proof in Supplementary A.1. +■ +To further make the estimation converge faster, we use non- +backtracking random walk. Given the start node 𝑠 and walk length +𝐿, its function is defined as follows: +W(𝑠, 𝐿) = +� +(𝑤0 = 𝑠, ...,𝑤𝐿) +𝑤𝑙 ∈ 𝑁 (𝑤𝑙−1), ∀𝑙 ∈ [1, 𝐿] +𝑤𝑙−1 ≠ 𝑤𝑙+1, ∀𝑙 ∈ [1, 𝐿 − 1] , +(3) +Data: Adjacency matrix 𝑨, number of trials 𝑀, number +of steps 𝐿, and dimension 𝑑 +Result: Emphasis matrix 𝑨∗ +1 𝑾 ′ ← 𝑶𝑛×𝑛; +/* approximate proximity matrix by random walk +*/ +2 for node 𝑖 in 𝐺 do +3 +for 𝑚 = 1, ..., 𝑀 do +4 +for 𝑗 ∈ W(𝑖, 𝐿) do +5 +𝑾 ′ +𝑖𝑗 ← 𝑾 ′ +𝑖𝑗 + 1; +6 +end +7 +end +8 end +/* masking, degree normalization and logarithm +*/ +9 𝑾𝑛×𝑛 ← log (𝑫−1(𝑾 ′ ◦ 𝑨)); +// proximity matrix +10 𝑼𝑛×𝑑, 𝚺𝑑×𝑑, 𝑽𝑇 +𝑑×𝑛 ← SVD(𝑾,𝑑); +// embedding +11 Weigh 𝑨∗ +𝑛×𝑛, where 𝑨∗ +𝑖𝑗 = S(𝑼𝑖, 𝑼 𝑗), ∀{𝑖, 𝑗|𝑨𝑖𝑗 = 1}; +12 Return 𝑨∗; +Algorithm 3: “Emphasis” Matrix +where 𝑁 (𝑖) denotes the neighbors of node 𝑖. Thus, with the same +𝐿 and 𝑀, we improve Lemma 1 to have a tighter bound of 𝜖: +LEMMA 2 (CONVERGENCE OF NON-BACKTRACKING RAN- +DOM WALKS). With the same condition as in Lemma 1, the error +𝜖 by a non-backtracking random walks is less than +𝜖 ≤ ⌈(𝐿 − 1)/3⌉ +𝐿 +√︂ +log (2/𝛿) +2𝐿𝑀 +(4) +PROOF. Omitted for brevity. Proof in Supplementary A.1. +■ +For example, when using regular random walks of length +𝐿 = 4 with 𝑀 = 30 trials, the estimated error by Lemma 1 with +probability 95% is about 6.2%. Nevertheless, if we instead use +non-backtracking random walks, the error is reduced to 3.1%, +which is 2× lower than the one by regular walks, indicating that +the approximated distribution converges well to the true one. +In Algorithm 3 line 9, an element-wise multiplication by 𝑨 +is done to keep the approximation of 1-hop neighbor for each +node, which sufficiently supplies necessary information as well +as keeps the resulting matrix sparse. We use the inverse of the +degree matrix 𝑫−1 to reduce the influence of nodes with large de- +grees. This prevents them from dominating the pairwise distance +by containing more elements in their rows. The element-wise +logarithm aims to rescale the distribution in 𝑾, in order to en- +large the difference between smaller structures. We use SVD for +efficient rank-𝑑 decomposition of the sparse proximity matrix +𝑾. We multiply the left-singular vectors 𝑼 by the corresponding +squared eigenvalues +√ +𝚺 to correct the scale. +Node Similarity. To estimate the node similarity, we compute +the distance of nodes in the embedding space. The intuition is that +the nodes that are closer in the embedding space should be better +connected with higher-order structures. Given the aforementioned +embedding 𝑼, the node similarity function S is: +S(𝑼𝑖, 𝑼 𝑗) = 𝑒−D(𝑼 𝑖𝑘,𝑼 𝑗𝑘), +(5) +where 𝑒 ≈ 2.718 denotes Euler’s number. Equation 5 is a universal +law proposed by Shepard [32], connecting the similarity with +distance via an exponential function. While the function D can + +Under Submission, , +Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos +be any distance metric, we use Euclidean because it is empirically +shown to work well. Negative exponential distribution is used +to bound the similarity from 0 to 1, which is close to 0 if the +distance is too large. Given 𝑨 and 𝑼, “Emphasis” Matrix 𝑨∗ +with weighted edges estimated by S is defined in line 11. Since +S(𝑼𝑖, 𝑼 𝑗) = S(𝑼 𝑗, 𝑼𝑖), 𝑨∗ is still a symmetric matrix. This is a +convenient property, which is later used for the fast computation +of the spectral radius (see Lemma 4). +4.2 +Compatibility Matrix Estimation +A compatibility matrix contains the class-wise strength of edges +and is important for properly inferring the node labels. In this +subsection, we show how to turn compatibility matrix estimation +into an optimization problem by introducing our closed-form +formula, which overcomes the defect of edge counting. We then +illustrate how we conquer several practical challenges to give a +precise and fast estimation. +Why NOT Edge Counting. The naive way to estimate com- +patibility matrix is via counting labeled edges. However, it is +inaccurate and has limitations: 1) rare labels will get neglected, +and 2) being noisy or biased due to few labeled nodes in real +graphs. The result is even more unreliable if the given labels are +imbalanced. Figure 3 is an example that edge counting fails if we +upsample 10× labels for only class 1. This occurs commonly in +practice, since we have only partial labels in node classification +tasks, and becomes fatal if the observed distribution is different +from the true one. +Closed-Form Formula. In Equation 1, if we initialize the final +belief with the initial one, and omit the addition of the initial +belief for the iterative propagation purpose, we have: +ˆ𝑩 = 𝑨ˆ𝑬 ˆ𝑯 +(6) +Our goal is to estimate the compatibility matrix ˆ𝑯 of a given +graph, so that the difference between belief propagated by the +given priors and the final belief is minimized. To solve this, we +firstly derive the closed-form solution of Equation 6 based on our +proposed Network Effect Formula: +LEMMA 3 (NETWORK EFFECT FORMULA). Given adja- +cency matrix 𝑨 and initial and final beliefs ˆ𝑬 and ˆ𝑩, the closed- +form solution of vectorized compatibility matrix vec( ˆ𝑯) is: +vec( ˆ𝑯) = (𝑿𝑇 𝑿)−1𝑿𝑇𝒚, +(7) +where 𝑿 = 𝑰𝑐×𝑐 ⊗ (𝑨ˆ𝑬) and 𝒚 = vec( ˆ𝑩). +PROOF. Omitted for brevity. Proof in Supplementary A.2. +■ +Although the final belief matrix ˆ𝑩 is not available before we +run actual propagation on the graph, we can replace it by 𝒚 = +vec( ˆ𝑬), and extract the ones that are corresponding to the priors +P. In other words, we change the problem into minimizing the +difference between initial belief of each node 𝑖 ∈ P by the initial +beliefs of its neighbors in the priors P, i.e., 𝑁 (𝑖) ∩ P. Intuitively, +neighbors should be able to estimate the belief for the node. The +optimization problem can then be formulated as follows: +min +ˆ𝑯 +∑︁ +𝑖 ∈P +𝑐∑︁ +𝑢=1 +ˆ𝑬𝑖𝑢 − ( +𝑐∑︁ +𝑘=1 +∑︁ +𝑗 ∈𝑁 (𝑖)∩P +ˆ𝑬 𝑗𝑘 ˆ𝑯𝑘𝑙) +(8) +With the help of Network Effect Formula, the optimization prob- +lem can then be solved by regression. +1 +2 +3 +4 +5 +6 +Class ID +1 +2 +3 +4 +5 +6 +Class ID +Edge Counting +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(a) Balanced Prior +1 +2 +3 +4 +5 +6 +Class ID +1 +2 +3 +4 +5 +6 +Class ID +Edge Counting +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(b) Imbalanced Prior +Figure 3: +Edge counting can not handle imbalanced case. +Class 1 is upsampled in this example. +Data: Emphasis Matrix 𝑨∗, initial belief ˆ𝑬, and priors P +Result: Estimated compatibility matrix ˆ𝑯∗ +1 𝒊 ← ∅; +// indices only related to priors +2 for 𝑝 ∈ P do +3 +for 𝑗 = 1, ...,𝑐 do +4 +𝒊 ← 𝒊 ∪ {𝑝 + (𝑗 − 1) ∗ 𝑐}; +5 +end +6 end +7 𝑿 ← (𝑰𝑐×𝑐 ⊗ (𝑨∗ ˆ𝑬)); +// feature matrix +8 𝒚 ← vec( ˆ𝑬); +// target vector +9 ˆ𝑯∗ ← 𝑅𝑖𝑑𝑔𝑒𝐶𝑉 (𝑿 [𝒊],𝒚[𝒊]); +10 Return row-normalize(max ( ˆ𝑯∗, 0)); +Algorithm 4: Compatibility Matrix Estimation +Practical Challenges and Solutions. Network Effect Formula +allows us to estimate the compatibility matrix by solving this op- +timization problem, but there still exists two practical challenges +that need to be addressed. +First, with few labels, it is difficult to properly separate them +into training and validation sets for the regression. We thus use +ridge regression with leave-one-out cross-validation (RidgeCV) +instead of the traditional linear regression. This allows us to fully +exploit the observations without having a bias caused by random +splits of training and validation sets. Moreover, the regularization +effect of ridge regression makes the compatibility matrix more +robust to noisy observations. It is noteworthy that the additional +computational cost of RidgeCV is negligible. +Next, the compatibility matrix estimated with the adjacency +matrix 𝑨 is easily interfered with by noisy neighbors, i.e., weakly- +connected pairs. To address this issue, we use our proposed “Em- +phasis” Matrix 𝑨∗ instead (see Section 4.1), to pay attention to +the labels of neighbors that are structurally important. Since the +rows of the estimated matrix 𝑯 do not sum to one in this ap- +proach, we filter out the negative values and normalize the sum +of each row to one. This is done safely, since the negative values +represent negligible relationships between nodes. +Algorithm. The overall process of estimation is shown in Al- +gorithm 4. We extract the indices that are corresponding to the +priors after the Kronecker product and vectorization in line 2 to +7. The optimization is then conducted in line 8 to 10 to estimate +the compatibility matrix ˆ𝑯∗. The negative value filtering and row +normalization is done on line 11. + +ULTRAPROP: Principled and Explainable Propagation on Large Graphs +Under Submission, , +4.3 +Theoretical Analysis +Convergence Guarantee. To ensure the convergence of propa- +gation, we introduce a scaling factor multiplied to it during the +iterations. The exact convergence of ULTRAPROP is as follows: +LEMMA 4 (EXACT CONVERGENCE). The criterion for the +exact convergence of ULTRAPROP is: +ULTRAPROP exactly converges ⇔ 0 < 𝑓 < +1 +𝜌(𝑨∗) , +(9) +where 𝜌(·) denotes the spectral radius of the given matrix. +PROOF. Omitted for brevity. Proof in Supplementary A.3. +■ +A smaller scaling factor leads to a faster convergence, never- +theless, distorts the results. In ULTRAPROP, we recommend a +large eigenvalue close to 1, setting 𝑓 = 0.9/𝜌(𝑨∗) as a reason- +able default. Since 𝑨∗ is built to be symmetric and sparse (see +Section 4.1), the computation of the spectral radius can be done +efficiently. +Complexity Analysis. ULTRAPROP uses sparse matrix repre- +sentation of graphs. The time complexity is given as: +LEMMA 5. ULTRAPROP scales linearly on the input size. the +time complexity of ULTRAPROP is at most +𝑂(𝑚), +(10) +and the space complexity is at most +𝑂(max (𝑚,𝑛 · 𝐿 · 𝑀) + 𝑛 · 𝑐2). +(11) +PROOF. Omitted for brevity. Proof in Supplementary A.4. +■ +5 +EXPERIMENTS +In this section, we aims to answer the following questions. +Q1. Accuracy: How well does ULTRAPROP work on real-world +graphs as compared to the baselines? +Q2. Scalability: How does the running-time of ULTRAPROP +scale w.r.t. graph size? +Q3. Explainability: How to explain the results of ULTRAPROP? +Experimental Setup +Datasets. We focus on large graphs and include eight graph +datasets with at least 22.5K nodes (details in Supplementary B.1) +in our evaluation. The statistics of datasets are shown in Table 2 +and 3. For each dataset, we sample only a few node labels as +initial beliefs. We do this for five times and report the average +and standard deviation to omit the biases. +“Synthetic” is the enlarged version of the graph shown in +Figure 1, which contains both heterophily and homophily NE. +Noisy edges are injected in the background, and the dense blocks +are constructed by randomly generating higher-order structures. +Baselines. We compare ULTRAPROP with five state-of-the-art +baselines and separate them into four groups: General GNNs: +GCN [16], and APPNP [17]. Heterophily GNN: MIXHOP [2], +and GPR-GNN [6]. BP-based methods: HOLS [8]. Our pro- +posed methods: ULTRAPROP-Hom and ULTRAPROP. ULTRA- +PROP-Hom is ULTRAPROP using identity matrix as compatibility +matrix, which assumes homophily and does not handle NE. The +details of baselines are given in Supplementary B.2. +Experimental Settings. For deep graph models, since we fo- +cus on the graph without node features, the node degrees are +transformed into one hot encoding and used as the node fea- +tures, which is suggested and implemented by several studies +(e.g. GraphSAGE and PyTorch Geometric) [9, 12, 13]. The de- +tails of hyperparameters are given in Supplementary B.3. To give +fair comparisons on run time, all the experiments are run on the +same machine, which is a stock Linux server with 3.2GHz Intel +Xeon CPU. In Section 5.2, we further investigate how much the +extra cost is, if a more powerful and but more expensive machine +is used. +5.1 +Q1 - Accuracy +In Table 2 and 3, we report the accuracy and wall-clock time for +each method. We highlight the top three from dark to light by +, +and +denoting the first, second and third place. +OBSERVATION 3. ULTRAPROP wins on X-ophily, heterophily +and homophily datasets. +X-ophily and Heterophily. In Table 2, ULTRAPROP outper- +forms all the competitors significantly by more than 34.4% and +12.8% accuracy on the “Synthetic” and “Pokec-Gender” datasets, +respectively. These datasets have strong NE, thus ULTRAPROP +boosts the accuracy owing to precise estimations of compatibility +matrix. The success in “Synthetic” further demonstrates its ability +to handle the dataset with X-ophily. Heterophily GNNs, namely +MIXHOP and GPR-GNN, all fail to predict correctly, giving +results close to random guessing. With homophily assumption, +General GNNs and BP-based methods also perform poorly. +Both “arXiv-Year” and “Patent-Year” datasets are shown to +only have weak NE (in Section 3.2), thus resulting in relatively +low accuracy for all methods compared with the other two datasets +with strong NE. Even so, ULTRAPROP still outperforms the +competitors by estimating a reasonable compatibility matrix. In +“arXiv-Year”, ULTRAPROP receives the second place by running +74.6× faster than MIXHOP. In “Patent-Year”, only ULTRAPROP, +APPNP and MIXHOP are able to give accuracy higher than +random guessing, which is 26.1%. +In the cases that ULTRAPROP is faster than ULTRAPROP- +Hom is because of both the low cost of compatibility matrix +estimation, and the lower spectral radius of ˆ𝑯∗, leading to a +faster convergence while propagating. +Homophily. In Table 3, ULTRAPROP-Hom outperforms all +the competitors on two homophily datasets, namely “GitHub” +and “Pokec-Locality”. ULTRAPROP performs similarly to UL- +TRAPROP-Hom, indicating its generalizability to the homophily +datasets by estimating near-identity matrices. In addition, ULTRA- +PROP-Hom gives competitive results with HOLS on the other +two homophily datasets “Facebook” and “arXiv-Category”, while +being 84.9× and 5.7× faster than HOLS respectively. General +GNNs rely heavily on node features for inference which explains +their poor performance. +OBSERVATION 4. Our optimizations makes difference. +We evaluate the effect of different compatibility matrices – (i) +ULTRAPROP-EC conducts edge counting on the labels of adja- +cent nodes in the priors, instead of using our Network Effect +Formula, and (ii) ULTRAPROP-A uses the adjacency matrix in- +stead of “Emphasis” Matrix to estimate the compatibility matrix + +Under Submission, , +Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos +Table 2: ULTRAPROP wins on X-ophily and Heterophily datasets. Accuracy, running time, and speedup are reported. Win- +ners and runner-ups in +, +and +. +Dataset +Synthetic +Pokec-Gender +arXiv-Year +Patent-Year +# of Nodes / Edges / Classes +1.2M / 34.0M / 6 +1.6M / 22.3M / 2 +169K / 1.2M / 5 +1.3M / 4.3M / 5 +Label Fraction +4% +0.4% +4% +4% +NE Strength +Strong +Strong +Weak +Weak +NE Type +X-ophily +Heterophily +X-ophily +Heterophily +Method +Accuracy (%) +Time (s) +Speedup Accuracy (%) +Time (s) +Speedup Accuracy (%) +Time (s) +Speedup Accuracy (%) +Time (s) +Speedup +GCN +16.7±0.0 +3456 +4.7× +51.8±0.1 +2906 +3.9× +35.3±0.1 +132 +3.3× +26.0±0.0 +894 +3.3× +APPNP +18.6±1.1 +7705 +10.4× +50.9±0.3 +6770 +9.1× +33.5±0.2 +423 +10.6× +27.5±0.2 +2050 +7.6× +MIXHOP +16.7±0.0 +58391 +79.0× +53.4±1.2 +53871 +72.7× +39.6±0.1 +2983 +74.6× +26.8±0.1 +18787 +70.1× +GPR-GNN +18.9±1.2 +7637 +10.3× +50.7±0.2 +6699 +9.0× +30.1±1.4 +400 +10.0× +25.3±0.1 +2034 +7.6× +HOLS +46.1±0.1 +1672 +2.3× +54.4±0.1 +8552 +11.5× +34.1±0.3 +566 +14.2× +23.6±0.0 +510 +1.9× +ULTRAPROP-Hom +45.7±0.1 +726 +1.0× +56.9±0.1 +736 +1.0× +37.0±0.3 +44 +1.0× +24.1±0.0 +316 +1.2× +ULTRAPROP +80.5±0.0 +739 +1.0× +67.2±0.1 +742 +1.0× +38.9±0.3 +42 +1.0× +28.6±0.1 +268 +1.0× +Table 3: ULTRAPROP wins on Homophily datasets. Accuracy, running time, and speedup are reported. Winners and runner- +ups in +, +and +. +Dataset +Facebook +GitHub +arXiv-Category +Pokec-Locality +# of Nodes / Edges / Classes +22.5K / 171K / 4 +37.7K / 289K / 2 +169K / 1.2M / 40 +1.6M / 22.3M / 10 +Label Fraction +4% +4% +0.4% +0.4% +Method +Accuracy (%) +Time (s) +Speedup Accuracy (%) +Time (s) +Speedup Accuracy (%) +Time (s) +Speedup Accuracy (%) +Time (s) +Speedup +GCN +67.0±0.8 +12 +3.0× +81.0±0.6 +28 +2.5× +24.5±0.6 +209 +1.7× +17.3±0.4 +4002 +3.3× +APPNP +50.5±2.2 +46 +10.5× +74.2±0.0 +73 +6.6× +17.7±1.3 +993 +8.1× +16.8±1.7 +11885 +9.7× +MIXHOP +69.2±0.7 +296 +73.5× +77.8±1.3 +526 +47.8× +23.6±0.5 +3029 +24.8× +16.9±0.3 +52139 +43.9× +GPR-GNN +51.9±1.5 +47 +11.8× +74.1±0.1 +75 +6.8× +18.4±1.2 +1016 +8.3× +30.0±2.0 +11959 +9.7× +HOLS +86.0±0.4 +934 +84.9× +80.8±0.5 +126 +11.5× +52.0±0.5 +692 +5.7× +63.7±0.3 +8139 +6.6× +ULTRAPROP-Hom +84.7±0.5 +4 +1.0× +81.7±0.7 +11 +1.0× +49.5±1.2 +124 +1.0× +65.4±0.3 +1270 +1.0× +ULTRAPROP +84.7±0.5 +4 +1.0× +81.7±0.7 +11 +1.0× +48.4±2.5 +122 +1.0× +64.6±1.0 +1231 +1.0× +Table 4: Ablation Study: Estimating compatibility matrix by +the proposed “Emphasis” Matrix is essential. Accuracy (%) +is reported in the table. +Datasets +NE Strength +ULTRAPROP-Hom ULTRAPROP-EC ULTRAPROP-A +ULTRAPROP +Synthetic +Strong +77.7±0.0 +68.0±0.1 +77.4±0.0 +80.5±0.0 +Pokec-Gender +56.9±0.1 +64.9±0.2 +64.8±0.2 +67.2±0.1 +arXiv-Year (imba.) +Weak +37.0±0.3 +36.5±1.0 +35.7±0.6 +38.4±0.0 +Patent-Year (imba.) +24.1±0.0 +24.0±0.9 +28.7±0.1 +28.7±0.0 +Table 5: ULTRAPROP is thrifty. AWS total dollar amount ($) +is reported in the table. The blue and red fonts denote run- +ning a single experiment by t3.small and p3.2xlarge, respec- +tively. Accuracy (%) is reported in Table 2 and 3. +Datasets +ULTRAPROP +GCN +Pokec-Gender +$ 0.28 (1.0×) +$ 12.61 (45.0×) +Pokec-Locality +$ 0.47 (1.0×) +$ 13.66 (29.1×) +in Algorithm 4. To demonstrate effectiveness of our proposed +estimation over edge counting, we upsample 5% labels to the +class with the fewest labels in the datasets with weak NE, which +are class 2 in “arXiv-Year” and class 1 in “Patent-Year”. We use +the original labels for propagation in the imbalanced datasets. +In Table 4, we find that ULTRAPROP outperforms all its vari- +ants in four datasets. In the datasets with strong NE, ULTRAPROP +shows its robustness to the structural noises and gives better re- +sults. In the imbalanced datasets, while ULTRAPROP-EC brings +its vulnerability to light, ULTRAPROP stays with high accuracy. +This study highlights the importance of a precise compatibility +matrix estimation, as well as forming it into an optimization +problem by our Network Effect Formula as shown in Lemma 3. +Furthermore, we compare ULTRAPROP with LINBP to dis- +play its advantages in Figure 4. In Figure 4a, the accuracy gap +between them indicates the necessity of precisely estimating the +compatibility matrix. In figure 4b, owing to “Emphasis” Matrix, +ULTRAPROP-Hom improves the accuracy in all homophily cases +100 +101 +102 +103 +Run Time +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Accuracy +LinBP +UltraProp +Synthetic +Pokec-Gender +arXiv-Year +Patent-Year +Facebook +GitHub +arXiv-Category +Pokec-Locality +(a) Run Time vs. Accuracy +Synthetic +Pokec-Gender +arXiv-Year +Patent-Year +Facebook +GitHub +arXiv-Category +Pokec-Locality +0.2 +0.4 +0.6 +0.8 +Accruacy +LinBP +UltraProp-Hom +UltraProp +1.8x +(b) Accuracy +Figure 4: Ablation Study: ULTRAPROP wins. It provides the +best trade-off between accuracy and running time compared +with LINBP. +compared with LINBP; owing to both “Emphasis” Matrix and +Network Effect Formula, ULTRAPROP improves the accuracy +in all cases while adding negligible penalty on run time, provid- +ing the best trade-off compared with LINBP. ULTRAPROP per- +forming similarly to ULTRAPROP-Hom on homophily datasets, +indicates that it correctly estimates near-identity matrices. + +ULTRAPROP: Principled and Explainable Propagation on Large Graphs +Under Submission, , +5.2 +Q2 - Scalability +We vary the edge number in “Pokec-Gender” and plot against +the wall-clock running time for ULTRAPROP in Figure 1c, in- +cluding both training and inference time. As there is no good +way to sample the graph [19], and also it is prohibitive to use +graph generator with million nodes, we try our best to ensure the +connectivity by continuously removing the nodes in the graph, +until the number of edges is no greater than the target. Note that +ULTRAPROP scales linearly as expected from Lemma 5. +Not only ULTRAPROP is scalable and linear, but it is also +thrifty, achieving up to 45× savings in dollar cost. It requires only +CPU, while comparable speeds by competitors, require GPUs. +Table 5 shows the estimated cost, assuming that we use a small +CPU machine for ULTRAPROP, and a GPU machine for GCN. +Details of computation are provided in Supplementary B.4. +5.3 +Q3 - Explainability +OBSERVATION 5. ULTRAPROP estimated the correct compat- +ibility matrices. +We illustrate that the estimations of compatibility matrix by +Network Effect Formula are precise in Figure 5, so as to inter- +preting the interrelations of classes extremely well. The inter- +relations of shown estimated compatibility matrices are similar +to the ones of edge counting in Figure 2, while being more ro- +bust to the noisy neighbors, namely, weakly connected ones. For +“Synthetic”, ULTRAPROP gives the exact answer that we use +to generate the dataset. For “Pokec-Gender”, ULTRAPROP suc- +cessfully estimates that people tend to connect to the ones with +opposite gender. This corresponds to the fact that people incline +to have more opposite gender interactions during their reproduc- +tive age [11], where the average ages of male and female in the +dataset are 25.4 and 24.2, respectively. Although “arXiv-Year” +and “Patent-Year” do not have strong NE, ULTRAPROP still gives +an estimated compatibility matrices making much sense in the +real world, where the papers and patents only cite to the ones +whose published dates are relatively close to them. We omit the re- +sults on homophily datasets, for brevity. In all cases ULTRAPROP +resulted in an near-identity compatibility matrix, as expected, +supported by giving similar results as ULTRAPROP-Hom, which +uses identity matrix as compatibility matrix. +6 +CONCLUSIONS +We firstly presented Network Effect Analysis (NEA) to identify +whether a graph exhibit network-effect or not, and surprisingly dis- +cover the absence of it in many real-world graphs known to have +heterophily. Next, we present ULTRAPROP to solve node classi- +fication based two insights, network-effect (NE) and neighbor- +differentiation (ND), which has the following advantages: +(1) Accurate: thanks to the precise compatibility matrix esti- +mation by NE, and ND that weighs important neighbors. +(2) Explainable: it interprets interrelations of classes with the +estimated compatibility matrix. +(3) Scalable: it scales linearly with the input size. +(4) Principled: it provides provable guarantees (Lemma 1, 2 +and 4) and closed-form solution (Lemma 3). +Applied on real-world million-scale graph datasets with over +22M edges, ULTRAPROP only requires 12 minutes on a stock +1 +2 +3 +4 +5 +6 +Class ID +1 +2 +3 +4 +5 +6 +Class ID +Est. Comp. Matrix +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(a) +“Synthetic”: X-ophily with +Strong NE +1 +2 +Class ID +1 +2 +Class ID +Est. Comp. Matrix +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(b) “Pokec-Gender”: Heterophily +with Strong NE +1 +2 +3 +4 +5 +Class ID +1 +2 +3 +4 +5 +Class ID +Est. Comp. Matrix +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +(c) +“arXiv-Year”: X-ophily with +Weak NE +1 +2 +3 +4 +5 +Class ID +1 +2 +3 +4 +5 +Class ID +Est. Comp. Matrix +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +(d) +“Patent-Year”: Heterophily +with Weak NE +Figure 5: ULTRAPROP is explainable. The estimated com- +patibility matrices are similar to the edge counting matrix +(in Figure 2), while being robust to the noises. +CPU-machine, and outperforms recent baselines on accuracy, as +well as on speed (≥ 9×). +Reproducibility: Our implemented source code and prepro- +cessed datasets will be published once the paper is accepted. + +Under Submission, , +Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos +REFERENCES +[1] Nvidia rtx a6000 deep learning benchmarks. https://lambdalabs.com/blog/ +nvidia-rtx-a6000-benchmarks/. +[2] S. Abu-El-Haija, B. 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We define the random variable 𝑋, +denoting the probability of node 𝑖 will walk to its 𝑗-th neighbor: +𝑋 = P(node 𝑖 walks to 𝑁 (𝑖)𝑗) = +�|𝑆 | +𝑘=1 1(𝑁 (𝑖)𝑗 = 𝑆𝑘) +|𝑆| +, +(12) +where P denotes the probability and 1 denotes the indicator. With +regular random walk in the graph without self-loops, the random +variable 𝑋 is upper-bounded by ⌈(𝐿−1)/2⌉ +𝐿 +. We can thus apply +Hoeffding’s inequality: +P(| ˆ𝜇|𝑆 | − 𝜇| ≥ 𝜖) ≤ 2 exp +−2𝐿3𝑀𝑡2 +⌈(𝐿 − 1)/2⌉2 , +(13) +where ˆ𝜇|𝑆 | denotes the sampled mean of the given random vari- +able, and 𝜇 denotes the expectation. Let 𝛿 = 2 exp +−2𝐿3𝑀𝑡2 +⌈(𝐿−1)/2⌉2 , +with probability 1 − 𝛿, the error 𝜖 is: +𝜖 = | ˆ𝜇|𝑆 | − 𝜇| ≤ ⌈(𝐿 − 1)/2⌉ +𝐿 +√︂ +log (2/𝛿) +2𝐿𝑀 +(14) +With the help of non-backtracking random walk [3], we can +further shrink the upper bound of 𝑋 into ⌈(𝐿−1)/3⌉ +𝐿 +. Now, let +𝛿 = 2 exp +−2𝐿3𝑀𝑡2 +⌈(𝐿−1)/3⌉2 , with probability 1 − 𝛿, the error 𝜖 can thus +be improved to: +𝜖 = | ˆ𝜇|𝑆 | − 𝜇| ≤ ⌈(𝐿 − 1)/3⌉ +𝐿 +√︂ +log (2/𝛿) +2𝐿𝑀 +(15) +■ +A.2 +Proof of Lemma 3 +PROOF. In the beginning, we introduce two necessary nota- +tions. vec(·) denotes the vectorization operator: +vec(𝑿) = [𝑿11, · · · , 𝑿𝑚1, 𝑿12, · · · , 𝑿𝑚2, · · · , 𝑿𝑚𝑛]⊤, +(16) +where 𝑿 is an 𝑚 ×𝑛 matrix, and 𝑿𝑖𝑗 denotes the element of 𝑿 on +the 𝑖-th row and the 𝑗-th column. Next, the Knronecker product +of given two 𝑚 × 𝑛 matrices 𝑿 and 𝒀 is: +𝑿 ⊗ 𝒀 = + +𝑿11𝒀 +𝑿12𝒀 +· · · +𝑿1𝑛𝒀 +𝑿21𝒀 +𝑿22𝒀 +· · · +𝑿2𝑛𝒀 +... +... +... +... +𝑿𝑚1𝒀 +𝑿𝑚2𝒀 +· · · +𝑿𝑚𝑛𝒀 + +(17) +The idea of this proof is to reformulate the equation in order to +derive the final result by the closed formula of Linear Regression. +We firstly show two well-known equations that will be used in +our proof. Given the features 𝑿 and target 𝒚, the closed formula +of the weights 𝑾 of Linear Regression is: +𝑾 = (𝑿𝑇 𝑿)−1𝑿𝑇𝒚. +(18) +The famous property of the mixed Kronecker matrix-vector prod- +uct [4] is also used: +vec(𝑩𝑽𝑨𝑇 ) = (𝑨 ⊗ 𝑩)𝒗, +(19) +where the matrix 𝑽 = vec−1(𝒗) is the result of the inverse of the +vectorization operator on 𝒗. +To begin the derivation, we vectorize Equation 6 into: +vec( ˆ𝑩) = vec((𝑨ˆ𝑬) ˆ𝑯𝑰𝑐×𝑐), +(20) +where 𝑰𝑐×𝑐 is a 𝑐 × 𝑐 identity matrix. The trick here, which is the +key of this proof, is to multiply one more identity matrix by ˆ𝑯. +Therefore, we use Equation 19 to reformulate the equation to: +vec( ˆ𝑩) = (𝑰𝑐×𝑐 ⊗ (𝑨ˆ𝑬))vec( ˆ𝑯) +(21) +By letting 𝑿 = 𝑰𝑐×𝑐 ⊗ (𝑨ˆ𝑬) and 𝒚 = vec( ˆ𝑩) in Equation 18, we +can then derive the closed-form solution of vectorized compati- +bility matrix as follows: +vec( ˆ𝑯) = (𝑿𝑇 𝑿)−1𝑿𝑇𝒚 +(22) +■ +A.3 +Proof of Lemma 4 +PROOF. ULTRAPROP exactly converges if and only if +𝜌(𝑨∗)𝜌( ˆ𝑯∗) < 1. However, the compatibility matrix 𝑯∗ is row- +normalized, so the largest eigenvalue 𝜌(𝑯∗) = 1 is a constant, +and is less than one after centering. Thus, the scaling factor 𝑓 +multiplied to the propagation (in Algorithm 2 line 5) should be in +the range of (0, +1 +𝜌 (𝑨∗) ) to meet the criterion of exact convergence. +■ +A.4 +Proof of Lemma 5 +PROOF. In the neighbor-differentiation phase, for each ran- +dom walk, each node visits at most 𝐿·𝑀 unique nodes, so the max- +imum number of non-zero elements in 𝑾 is either 𝑛 · 𝐿 · 𝑀 if we +have not walked through all the edges, or 𝑚 otherwise. The time +complexity of SVD on 𝑾 then takes 𝑂(𝑑 · max (𝑚,𝑛 · 𝐿 · 𝑀)). In +the network-effect phase, the time complexity for the Fisher’s ex- +act test is 𝑂(max (𝑪)), where max (𝑪) is a constant bounded by +500 in our algorithm. Therefore, network-effect analysis takes +𝑂(|𝒆 +′| · 𝑐2). For the regression, since there are 𝑐 sets of pa- +rameters are independent, we can separate the problem into 𝑐 +tasks, where each contains 𝑐 features and |𝒑| samples. Thus +the complexity can be reduced to 𝑂(|𝒑| · 𝑐3), and the efficient +leave-one-out cross-validation only needs to be done once. In the +propagation phase, it takes at most 𝑂(𝑚 + 𝑛) for sparse matrix +multiplication to run 𝑡 iterations. Thus, the time complexity is +𝑂(𝑑 max (𝑚,𝑛 · 𝐿 · 𝑀) + |𝒑| ·𝑐3 +𝑚). However, in practice, 𝑐, |𝒑| +and 𝑡 are usually small constants which are negligible, and 𝑚 +is usually much larger than them. Therefore, keeping only the +dominating terms, the time complexity is approximately 𝑂(𝑚). +𝑾 contains at most max (𝑚,𝑛 · 𝐿 · 𝑀) non-zero elements. The +Kronecker product at most contains 𝑛 · 𝑐2 non-zero elements. ˆ𝑩 +and ˆ𝑯 contain at most 𝑛 ·𝑐 and 𝑐2 non-zero elements, respectively. +Thus, the space complexity is 𝑂(max (𝑚,𝑛 · 𝐿 · 𝑀) + 𝑛 · 𝑐2). +■ +B +REPRODUCIBILITY +B.1 +Datasets +• “Pokec-Gender” [33] is an online social network in Slo- +vakia. [23] re-labels the nodes by users’ genders instead. +• “arXiv-Year” [14] is a citation network between all Com- +puter Science arXiv papers. [23] re-labels the nodes by the +posted years. +• “Patent-Year” [20] is the patent citation network from +1980 to 1985. [23] re-labels the nodes by the application +year, bucketized into five consecutive 3-year ranges. +• “Synthetic” is a graph enlarged by the one in Figure 1. It +contains both heterophily and homophily network-effect. + +Under Submission, , +Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos +Table 6: Hyperparameters for Deep Graph Models +Method +Hyperparameters +GCN +lr=0.01, wd=0.0005, hidden=16, dropout=0.5 +APPNP +lr=0.002, wd=0.0005, hidden=64, dropout=0.5, K=10, alpha=0.1 +MIXHOP +lr=0.01, wd=0.0005, cutoff=0.1, layers1=[200, 200, 200], layers2=[200, 200, 200] +GPR-GNN +lr=0.002, wd=0.0005, hidden=64, dropout=0.5, K=10, alpha=0.1 +Noisy edges are randomly injected in the background, and +the dense blocks are constructed by randomly creating +higher-order structures. +• “Facebook” [30] is a page-to-page network of verified +Facebook sites. Nodes are labeled by the categories such +as politicians and companies. +• “GitHub” [30] is a social network of developers in June +2019. Nodes are labeled as web or a machine learning +developer. +• “arXiv-Category” [37] is the same dataset as the arXiv- +Year dataset. Nodes are labeled by the primary categories. +• “Pokec-Locality” [33] is the same dataset as the Pokec- +Gender dataset. Nodes are labeled by the uses’ localities. +B.2 +Baselines +• GCN2 [16] is a well-known deep graph model, learning +and aggregating the weights of two-hop neighbors. +• APPNP4 [17] utilizes personalized PageRank to leverage +the local information and a larger neighborhood. +• MIXHOP3 [2] mixes powers of the adjacency matrix to +incorporate more than 1-hop neighbors in each layer. +• GPR-GNN4 [6] allows the learnable weights to be nega- +tive during propagation with Generalized PageRank. +• HOLS5 [8] is a label propagation method with attention, +by increasing the importance of a neighbor if they appear +in the same motif at the same time. +B.3 +Hyperparameters +For ULTRAPROP and ULTRAPROP-Hom, we use random walks +of length 4 with 10 trials except GitHub, arXiv-Category and +Pokec-Locality datasets, where we use 30 trials. The decompo- +sition rank is set to be 128, which is empirically shown to be +enough in the embedding tasks. The weights of HOLS for differ- +ent motifs are set to be equal. For the deep graph models, under +the setting that the given labels are very few, it is impossible to +separate a validation set. We then train them for a fixed number +of epochs (i.e. 200 epochs), which is usually sufficient enough for +them to converge. All the fully connected layers are replaced by +the sparse version in order to fit into memory. Both adjacency ma- +trices and features are normalized and turn into sparse matrices if +needed. For other hyperparameters, we use the default settings +given by the authors, and give the details in Table 6. +2https://github.com/tkipf/pygcn +3https://github.com/benedekrozemberczki/MixHop-and-N-GCN +4https://github.com/jianhao2016/GPRGNN +5https://github.com/dhivyaeswaran/hols +B.4 +Scalability +We select machines provided by AWS with comparable specs as +we use for the experiments. For CPU machine, we select t3.small +with 3.3GHz CPU and 2GB RAM, which is faster than ours, and +costs $0.023 per hour. For GPU machine, we select p3.2xlarge +with a V100 GPU, which costs $3.06 per hour. According to [1], +it is 0.89 slower than the RTX A6000 GPU we use on running +PyTorch. The running time of GCN on “Pokec-Gender” and +“Pokec-Locality” are 673 and 730 seconds, respectively. Using the +provided information, the results in Table 5 can be computed. + diff --git a/2NAyT4oBgHgl3EQfbve8/content/tmp_files/load_file.txt b/2NAyT4oBgHgl3EQfbve8/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cf73f6734ba89a3eed6afbf7ccadddf7cd6775a2 --- /dev/null +++ b/2NAyT4oBgHgl3EQfbve8/content/tmp_files/load_file.txt @@ -0,0 +1,1286 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf,len=1285 +page_content='ULTRAPROP: Principled and Explainable Propagation on Large Graphs Meng-Chieh Lee mengchil@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='edu Pittsburgh, USA Carnegie Mellon University Shubhranshu Shekhar shubhras@andrew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='edu Pittsburgh, USA Carnegie Mellon University Jaemin Yoo jaeminyoo@cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='edu Pittsburgh, USA Carnegie Mellon University Christos Faloutsos christos@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='edu Pittsburgh, USA Carnegie Mellon University ABSTRACT Given a large graph with few node labels, how can we (a) identify the mixed network-effect of the graph and (b) predict the unknown labels accurately and efficiently?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' This work proposes Network Effect Analysis (NEA) and ULTRAPROP, which are based on two insights: (a) the network-effect (NE) insight: a graph can exhibit not only one of homophily and heterophily, but also both or none in a label-wise manner, and (b) the neighbor-differentiation (ND) insight: neighbors have different degrees of influence on the target node based on the strength of connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' NEA provides a statistical test to check whether a graph ex- hibits network-effect or not, and surprisingly discovers the ab- sence of NE in many real-world graphs known to have heterophily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ULTRAPROP solves the node classification problem with notable advantages: (a) Accurate, thanks to the network-effect (NE) and neighbor-differentiation (ND) insights;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (b) Explainable, precisely estimating the compatibility matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (c) Scalable, being linear with the input size and handling graphs with millions of nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' and (d) Principled, with closed-form formula and theoretical guar- antee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Applied on eight real-world graph datasets, ULTRAPROP outperforms top competitors in terms of accuracy and run time, requiring only stock CPU servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' On a large real-world graph with 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='6M nodes and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3M edges, ULTRAPROP achieves ≥ 9× speedup (12 minutes vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 2 hours) compared to most competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ACM Reference Format: Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Falout- sos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ULTRAPROP: Principled and Explainable Propagation on Large Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In Under Submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ACM, New York, NY, USA, 12 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='nnnnnnn 1 INTRODUCTION Given a large, undirected, and unweighted graph with few la- beled nodes, how can we infer the labels of remaining unlabeled nodes, often without node features?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Node classification is often employed to infer labels on large real-world graphs, since manual labeling is expensive and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' For example, in social networks with millions of users, identifying even a fraction (say 5%) of users’ groups is prohibitive, which limits the application of methods that assume a large fraction of labels are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' More- over, node features are frequently missing in real-world graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' For those methods that require node features in classification, Under Submission, , 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ACM ISBN 978-x-xxxx-xxxx-x/YY/MM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='00 https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='nnnnnnn they create the features based on the graph [9, 12, 13], such as using the one-hot encoding of node degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Previous works on node classification have two main limita- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' First, they ignore the complex network-effect of real-world graphs and understand their characteristic as either homophily or heterophily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The co-existing case of homophily and heterophily, which we call X-ophily in this work, has been neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Sec- ond, they either a) ignore the different influences of neighboring nodes during inference or b) require extensive computation to give dynamic weights to the adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In this work, we address these two challenges and consider the dynamic and com- plex relationships between neighboring nodes with two insights network-effect and neighbor-differentiation for designing an ac- curate and efficient approach for node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' NE (network-effect): The first goal is to analyze the network- effect of a graph (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=', homophily, heterophily, or any combination which we call X-ophily) in a principled and class-conditional way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' That is, a single graph can have homophily and heterophily at the same time between different pairs of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The challenge is usually avoided in literature: inference-based methods assume that the relationship is given by domain experts [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' deep graph models either assume homophily [16, 39] or misidentify graphs having no NE as heterophily graphs [23, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ND (neighbor-differentiation): The second goal is to approx- imate different influence levels of neighboring nodes effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Existing works require extensive computation to measure the influence levels in node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' For instance, HOLS [8] solves ND by mining 𝑘−cliques, while listing all the instances is time-consuming;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Graph Attention Network (GAT) [35] learns more than one relationship for each neighbor, while heavily rely- ing on the node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We provide an informal definition of the problem: INFORMAL PROBLEM 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Given an undirected and unweighted graph – with few labeled nodes, – without node features, Infer the labels of all the remaining nodes – accurately under any types of network effects, – explaining the predictions to human experts, – efficiently in large-scale graphs with scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Our solutions: We propose Network Effect Analysis (NEA), an algorithm to statistically test NE of a real-world graph with only a few observed node labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' NEA analyzes the relationships between all pairs of different classes in an efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='00270v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='SI] 31 Dec 2022 Under Submission, , Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos Label 2400x6 Node ID Label ID Adjacency 2400x2400 Input Estimated Compatibility Matrix Homophily Heterophily Output Compatibility 6x6 (a) NE: Compatibility Matrix Estimation R3 R1 R2 X B4 B3 B2 B1 B8 B7 B6 B5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='66 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='96 Proposed Embedding Space Predicts Blue predicts Red ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' B1-4 R1-3 X (b) ND: Neighbor Differentiation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 # of Edges 1e7 0 2000 4000 6000 Run Time (s) UltraProp GCN HOLS 11x 4x (c) Scalability Figure 1: ULTRAPROP is Effective, Explainable, and Scalable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (a) Thanks to Network Effect Formula, ULTRAPROP ex- plains the dataset by precisely estimating the compatibility matrix, observing both heterophily and homophily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (b) Thanks to “Emphasis” Matrix, ULTRAPROP predicts the label of the gray node X correctly, while LINBP fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (c) ULTRAPROP is fast and scales linearly with the number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' See Introduction for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Figure 2, we show that surprisingly many large public datasets known as heterophily graphs do not have NE at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We then propose ULTRAPROP, a principled approach using both insights of NE and ND to conduct accurate node classifi- cation on large graphs with explainability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The explainability is built upon the combination of influential neighbors (ND) and the compatibility matrix that we carefully and automatically estimate (see Lemma 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Figure 1 illustrates the advantages of ULTRA- PROP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Figure 1a shows how ULTRAPROP provides explanation by estimating a compatibility matrix from only 5% of node labels: the interrelations of classes imply that the first half follows het- erophily, while the other half follows homophily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Figure 1b shows that ULTRAPROP predicts the different influences of neighbors correctly by ND, where the central vertex X is closer to the red nodes R1, R2, and R3 in the embedding space, as it participates in a closely-knit community with them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Finally, Figure 1c shows the linear scalability of ULTRAPROP with the number of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' It is 9× faster than most of the competitors, and requires only 12 minutes on a large real-world graph with over 22M edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In summary,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' the advantages of ULTRAPROP are (1) Accurate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' thanks to the precise estimation of the compati- bility matrix,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' and the reliable measurement of the different importance of neighbors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (2) Explainable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' interpreting the datasets with estimated com- patibility matrices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' which work for homophily,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' heterophily,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' or any combination – X-ophily,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (3) Scalable,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' scaling linearly with the input size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (4) Principled,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' providing a tight bound of convergence for the random walks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' and the closed-form formula for the compatibility matrix (see Lemma 2 and 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Reproducibility: Our implemented source code and prepro- cessed datasets will be published once the paper is accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 2 BACKGROUND AND RELATED WORK We introduce preliminaries, and related works on label propaga- tion and node embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Table 1 presents qualitative compari- son of state-of-the-art approaches against our proposed method ULTRAPROP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' No competitor fulfills all the specs in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Let 𝐺 be an undirected and unweighted graph with 𝑛 nodes and 𝑚 edges with 𝑨 as the adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 𝑨𝑖𝑗 = 1 indicates that nodes 𝑖, 𝑗 are connected by an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Each node 𝑖 has a unique label 𝑙(𝑖) ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ,𝑐}, where 𝑐 denotes the number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Let 𝑬 ∈ R𝑛×𝑐 be the initial belief matrix containing the prior information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=', the labeled nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 𝑬𝑖𝑘 = 1 if 𝑙(𝑖) = 𝑘, and the rest entries of the 𝑖𝑡ℎ row are filled up with zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' For the nodes without labels, all the entries corresponding to those nodes are set to 1/𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 𝑯 ∈ R𝑐×𝑐 is a row-normalized compatibility matrix where 𝑯𝑘𝑙 denotes the relative influence of class 𝑙 on class 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The residual of a matrix around 𝑘 is denoted as ˆ𝒀 and is defined as ˆ𝒀 = 𝒀 −𝑘 × 1 where 𝒀 is centered 1 around 𝑘, and 1 is matrix of ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 Label Propagation Belief Propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Belief Propagation (BP) is a popular method for label inference in graphs [10, 18, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' FABP [18] and LINBP [10] accelerate BP by approximating the final belief assignment from BP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In particular, LINBP approximates the final belief as: ˆ𝑩 = ˆ𝑬 + 𝑨ˆ𝑩 ˆ𝑯, (1) where ˆ𝑩 is a residual final belief matrix, initialized with all zeros, 𝑨 is the adjacency matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The compatibility matrix 𝑯 and initial beliefs 𝑬 are centered around 1/𝑐 to ensure convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Higher-Order Propagation Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' HOLS [8] leverages higher-order graph structures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 𝑘−cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' It propagates the labels by incorporating the weights from higher-order cliques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' However, mining cliques is computationally intensive, and pro- hibitive for large graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 Embedding Methods Traditional Embedding Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Numerous embedding meth- ods [5, 7, 29] have been proposed to capture neighborhood sim- ilarity and role of nodes in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' [5] propose a random walk based generalized embedding method to capture non-linear relations among nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Similarly, Pixie [7] utilizes localized random walk based on node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Further, [29] intro- duced a generalized method that derives the matrix closed forms of different graph embedding methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 1A matrix “centered around” 𝑘 has all its entries close to 𝑘 and the average of the entries is exactly 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ULTRAPROPLINBPULTRAPROPULTRAPROP: Principled and Explainable Propagation on Large Graphs Under Submission, , Table 1: ULTRAPROP matches all specs, while competitors miss one or more of the properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Each property corre- sponds to a contribution in Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ‘?’' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' indicates that it is unclear from the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Property Method BP [10, 18] HOLS [8] General GNNs [16, 17] Attention GNNs [15, 35] Heterophily GNNs [2, 6] ULTRAPROP Contr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (1): Handling NE ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' � Contr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (1): Handling ND � � � Contr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (2): Explainable � Contr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (3): Scalable � � � Contr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (4): Principled � � � Deep Graph Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Graph Convolutional Networks (GCN) [16] employ approximate spectral convolutions to incorporate neigh- borhood information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' APPNP [17] utilizes personalized PageR- ank to leverage the local information and a larger neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' To account for ND, Graph Attention Networks (GAT) [15, 35] al- low for assigning importance weights to neighborhoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' However, attention GNNs require node features, and need many learnable parameters, making it infeasible for large graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' MIXHOP [2] makes no assumption of homophily, and mixes powers of the adjacency matrix to incorporate more than 1-hop neighbors in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' H2GCN [41] is built on three key designs to better learn the structure of heterophily graphs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' nevertheless, it requires too much memory and thus is not able to handle large graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' GPR-GNN [6] allows the learnable weights to be negative during propagation with Generalized PageRank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' LINKX [23] introduces multiple large heterophily datasets, but it is not applicable to graphs without node features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' [26] empirically evaluates the per- formance of GNNs on small heterophily datasets (≤ 10K nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' However, most of the conclusions are made based on the evalua- tions where the node features are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' While deep graph models have been shown to be state-of-the-art methods, it relies on node features and is not scalable without GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Further, it is hard to supply explanations or provide theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 3 PROPOSED METHOD PART I – “NEA” Given a graph with few node labels, how can we identify what are the classes that a node with a specific class connects to?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In other words, how can we find whether the graph exhibits X-ophily – ho- mophily, heterophily, or even none?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We propose Network Effect Analysis (NEA), a statistical approach to identify the network- effect (NE) in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' It leads to interesting discovery that many widely used heterophily graphs exhibit no NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 Network Effect Analysis (NEA) Previous works on identifying NE of a graph [23, 41] have two main limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' First, when a class connects to all existing classes uniformly, they misunderstand this non-homophily class as heterophily, which should be considered as having no NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Sec- ond, they require the labels of most nodes in a graph, even though Data: Edges E and priors P Result: 𝑝-value table 𝑭 /* edges with both nodes in priors / 1 Extract E ′ such that (𝑖, 𝑗) ∈ E,𝑖, 𝑗 ∈ P ∀(𝑖, 𝑗) ∈ E ′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 2 𝑻 ← 𝑶𝑐×𝑐;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' // test statistic table /* do 𝜒2 test for 𝐵 times / 3 for 𝑏1 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=', 𝐵 do 4 for 𝑐1 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=',𝑐 do 5 for 𝑐2 = 𝑐1 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=',𝑐 do 6 𝑽 ← 𝑶2×2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' // contingency table 7 Shuffle(E ′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' // sampling 8 for (𝑖, 𝑗) ∈ E ′ do 9 if 𝑙(𝑖) = 𝑐1 and 𝑙(𝑗) = 𝑐1 then 10 𝑽11 ← 𝑽11 + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 11 else if (𝑙(𝑖) = 𝑐1 and 𝑙(𝑗) = 𝑐2) or 12 (𝑙(𝑖) = 𝑐2 and 𝑙(𝑗) = 𝑐1) then 13 𝑽21 ← 𝑽21 + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 14 𝑽12 ← 𝑽12 + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 15 else if 𝑙(𝑖) = 𝑐2 and 𝑙(𝑗) = 𝑐2 then 16 𝑽22 ← 𝑽22 + 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 17 if �2 𝑖=1 �2 𝑗=1 𝑽𝑖𝑗 > 250 then 18 Break;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 19 end 20 end /* record statistics of class pairs / 21 𝑇 = 𝜒2-Test-Statistic(𝑽);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 22 𝑻𝑐1𝑐2 ← 𝑻𝑐1𝑐2 +𝑇/𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 23 𝑻𝑐2𝑐1 ← 𝑻𝑐2𝑐1 +𝑇/𝐵;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 24 end 25 end 26 end 27 Compute 𝑝-value table 𝑭𝑐×𝑐 with average statistics in 𝑻;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 28 Return 𝑭;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Algorithm 1: Network Effect Analysis (NEA) in most real-world node classification tasks only a few node la- bels are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We propose NEA to address such limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Before introducing NEA, we provide two propositions: PROPOSITION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Given a graph and a class 𝑐𝑖, if the nodes with class 𝑐𝑖 tend to connect uniformly to the nodes with all classes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=',𝑐 equally, then class 𝑐𝑖 has no NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' PROPOSITION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' If all classes 𝑐𝑖 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=',𝑐 in a graph have no NE, then this graph has no NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We separate heterophily graphs from those with no NE by the propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In heterophily graphs, the nodes of a specific class are likely to be connected to the nodes of other classes, such as in bipartite graphs that connect different classes of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In this case, knowing the label of a node gives meaningful information about the labels about its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' On the other hand, if a graph has no NE, every node has equal probabilities for more than one class even after we consider the structural information from its neighbors, which is useless to infer its true label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' To analyze whether a specific class 𝑐𝑖 has NE or not, we use 𝜒2 test to identify whether there exists a statistically significant contingency between the classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Given two classes 𝑐1 and 𝑐2, the Under Submission, , Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos 1 2 Class ID 1 2 Class ID Edge Counting 105 106 2 × 105 3 × 105 4 × 105 6 × 105 1 2 Class ID 1 2 Class ID p-value Table 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='05 (a) “Genius”: No NE 1 2 Class ID 1 2 Class ID Edge Counting 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='6 × 105 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='7 × 105 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='8 × 105 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='9 × 105 7 × 105 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 × 105 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 × 105 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3 × 105 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4 × 105 1 2 Class ID 1 2 Class ID p-value Table 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='05 (b) “Penn94”: No NE 1 2 Class ID 1 2 Class ID Edge Counting 3 × 106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 × 106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4 × 106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='6 × 106 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='8 × 106 4 × 106 1 2 Class ID 1 2 Class ID p-value Table 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='05 (c) “Twitch”: No NE 1 2 3 4 5 Class ID 1 2 3 4 5 Class ID Edge Counting 105 3 × 104 4 × 104 6 × 104 2 × 105 1 2 3 4 5 Class ID 1 2 3 4 5 Class ID p-value Table 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='05 (d) “arXiv-Year”: X-ophily with Weak NE 1 2 3 4 5 Class ID 1 2 3 4 5 Class ID Edge Counting 105 2 × 105 3 × 105 4 × 105 6 × 105 1 2 3 4 5 Class ID 1 2 3 4 5 Class ID p-value Table 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='05 (e) “Patent-Year”: Heterophily with Weak NE 1 2 Class ID 1 2 Class ID Edge Counting 107 8 × 106 9 × 106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 × 107 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 × 107 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3 × 107 1 2 Class ID 1 2 Class ID p-value Table 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='05 (f) “Pokec-Gender”: Heterophily with Strong NE Figure 2: NEA discovers that real-world heterophily graphs do not necessarily have network-effect (NE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' For each dataset, we report the edge counting on the left, and the 𝑝-value table output from NEA on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We have a case of X-ophily, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' in “arXiv-Year”, class 1 is homophily, and the rest are heterophily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' input to the test is 2 × 2 contingency table with counts of edges where nodes of each edge ∈ {𝑐1,𝑐2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' NULL HYPOTHESIS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Edges are equally likely to exhibit homophily and heterophilly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Algorithm 1 presents the procedure for the proposed NEA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' A practical challenge is that if the numbers in the table are too large, 𝑝-value becomes extremely small and meaningless [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' However, sampling for only a single round can be unstable and output very different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' To address this, we combine 𝑝- values from different random sampling by Universal Inference [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We firstly sample edges to add to the contingency table until the frequency is above a specified threshold, and compute the 𝜒2 test statistic for each class pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Next, following Universal Inference, we repeat the procedure for random samples of edges for 𝐵 rounds and average the statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' At last, we use the average statistics to compute the 𝑝-value table 𝑭𝑐×𝑐 of 𝜒2 tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' It is worth noting that, NEA is robust to the noisy edges, thanks to the random sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' It also works well given either a few or many node labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Given only a few observations, 𝜒2 test works well enough when the frequency in the contingency table are only at least 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' given many observations, the sampling and combining trick ensures the correctness of 𝑝-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We give observations based on the result of NEA: OBSERVATION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' If a class accepts all the null hypotheses in Algorithm 1, then this class has no NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We then extend Observation 1 to an extreme case: OBSERVATION 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' If all classes in a graph obey Observation 1, the node classification problem is unsolvable under our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 Discoveries For each dataset, we equally sample 5% of node labels and com- pute the 𝑝-value table by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' This is because a) only a few labels are observed in most node classification tasks, and thus it is natural to make the same assumption in this analysis, and b) our NEA can correctly analyze NE even from partial ob- servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We set 𝐵 = 1000 to output stable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Based on Observation 2, here is our surprising discovery: DISCOVERY 1 (NO NE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' “Genius”, “Penn94”, and “Twitch” have no NE, exhibiting neither homophily nor heterophily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' “Genius” [22], “Penn94” [34], and “Twitch” [31] have been widely used in previous works [21, 23, 25, 27, 36, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In “Ge- nius” (Figure 2a), we see that both classes 1 and 2 tend to connect to class 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' This makes the class 2 indistinguishable by the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' NEA thus accepts the null hypothesis and identifies that there exists no statistically significant difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' This means that the edges have the same probabilities to be homophily and heterophily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We can see a similar phenomenon in “Penn94” (Fig- ure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' “Twitch” (Figure 2c) is not considered as a homophily graph because the effect is too weak, where the scales on the color bar are very close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' However, it is not a heterophily graph as well, where NEA correctly identifies that every class tends to connect to both classes near-uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We further analyzed three more datasets: DISCOVERY 2 (WEAK AND STRONG NE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' “Arxiv” and “Patent-Year” exhibit weak NE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' and “Pokec-Gender” exhibits strong NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The “arXiv-Year” and “Patent-Year” datasets (Figure 2d and 2e) have weak NE, where one of the classes accepts more than one null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' “Pokec-Gender” (Figure 2f) shows strong NE, where the estimated 𝑝-value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' These three datasets will later be used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 4 PROPOSED METHOD PART II – ULTRAPROP We propose ULTRAPROP, our approach for accurate node classifi- cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Algorithm 2 shows the algorithm of ULTRAPROP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In line 1, given an adjacency matrix 𝑨 and rank 𝑑, we make “Emphasis” Matrix 𝑨∗ (in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1) to handle the neighbor-differentiation (ND).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' To handle network-effect (NE), we estimate the compati- bility matrix ˆ𝑯∗ from 𝑨∗ in line 2 (in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In line 3 to 7, we initialize and propagate the beliefs ˆ𝑩 iteratively through 𝑨∗ until they converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In each iteration, we aggregate the beliefs of neighbors in ˆ𝑩, weighted by the values in 𝑨∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' This aims to draw attention to the neighbors that are more structurally important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ULTRAPROP: Principled and Explainable Propagation on Large Graphs Under Submission, , Data: Adjacency matrix 𝑨, initial belief ˆ𝑬, priors P, and decomposition rank 𝑑 Result: Final belief 𝑩 1 𝑨∗ ← “Emphasis”-Matrix(𝑨,𝑑);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 2 ˆ𝑯∗ ← Compatibility-Matrix-Estimation(𝑨∗, ˆ𝑬, P);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' /* propagation / 3 ˆ𝑩(0) ← 𝑶𝑛×𝑐,𝑡 ← 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 4 while inferences changed and � | ˆ𝑩(𝑡+1)− ˆ𝑩(𝑡) | 𝑛𝑐 > 1 lg𝑛𝑐 do 5 ˆ𝑩(𝑡+1) ← ˆ𝑬 + 𝑓 𝑨∗ ˆ𝑩(𝑡) ˆ𝑯∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 6 𝑡 ← 𝑡 + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 7 end 8 Return 𝑩 ← ˆ𝑩(𝑡) + 1 𝑐 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Algorithm 2: ULTRAPROP The interrelations between classes is handled by multiplying with ˆ𝑯∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We further include an early stopping criterion in line 4 for more efficient propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 “Emphasis” Matrix To incorporate the idea of ND, where neighbors have different importances, we propose to replace the unweighted adjacency matrix 𝑨 with a weighted one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The weight of edge (𝑖, 𝑗) reflects the influence of node 𝑖 for 𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We present an efficient solution to weigh 𝑨 without using any node labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' It firstly embeds nodes into structure-aware representations via random walks, and then measures their similarities via distances in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Structure-Aware Node Representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We represent nodes in 𝑑-dimensional vector space efficiently using Singular Value Decomposition (SVD) on the high-order proximity matrix of the graph and capture information from pairwise connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' To fast approximate the higher-order proximity matrix, we utilize random walks described in Algorithm 3 from line 1 to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Given a proximity matrix 𝑾 ′, 𝑾 ′ 𝑖𝑗 records the number of times we visit node 𝑗 if we start a random walk from node 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Each neighbor has the same probability of being visited in the unweighted graphs, where only those structurally important neighbors are visited more frequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' To theoretically justify why it works, we prove that the neigh- bor distribution for each node converges after a number of trials: LEMMA 1 (CONVERGENCE OF REGULAR RANDOM WALKS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' With probability 1−𝛿, the error 𝜖 between the approximated distri- bution and the true one for a node walking to its 1-hop neighbor by a regular random walk of length 𝐿 with 𝑀 trials is less than 𝜖 ≤ ⌈(𝐿 − 1)/2⌉ 𝐿 √︂ log (2/𝛿) 2𝐿𝑀 (2) PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Omitted for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Proof in Supplementary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ■ To further make the estimation converge faster, we use non- backtracking random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Given the start node 𝑠 and walk length 𝐿, its function is defined as follows: W(𝑠, 𝐿) = � (𝑤0 = 𝑠, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=',𝑤𝐿) 𝑤𝑙 ∈ 𝑁 (𝑤𝑙−1), ∀𝑙 ∈ [1, 𝐿] 𝑤𝑙−1 ≠ 𝑤𝑙+1, ∀𝑙 ∈ [1, 𝐿 − 1] , (3) Data: Adjacency matrix 𝑨, number of trials 𝑀, number of steps 𝐿, and dimension 𝑑 Result: Emphasis matrix 𝑨∗ 1 𝑾 ′ ← 𝑶𝑛×𝑛;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' /* approximate proximity matrix by random walk / 2 for node 𝑖 in 𝐺 do 3 for 𝑚 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=', 𝑀 do 4 for 𝑗 ∈ W(𝑖, 𝐿) do 5 𝑾 ′ 𝑖𝑗 ← 𝑾 ′ 𝑖𝑗 + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 6 end 7 end 8 end /* masking, degree normalization and logarithm / 9 𝑾𝑛×𝑛 ← log (𝑫−1(𝑾 ′ ◦ 𝑨));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' // proximity matrix 10 𝑼𝑛×𝑑, 𝚺𝑑×𝑑, 𝑽𝑇 𝑑×𝑛 ← SVD(𝑾,𝑑);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' // embedding 11 Weigh 𝑨∗ 𝑛×𝑛, where 𝑨∗ 𝑖𝑗 = S(𝑼𝑖, 𝑼 𝑗), ∀{𝑖, 𝑗|𝑨𝑖𝑗 = 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 12 Return 𝑨∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Algorithm 3: “Emphasis” Matrix where 𝑁 (𝑖) denotes the neighbors of node 𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Thus, with the same 𝐿 and 𝑀, we improve Lemma 1 to have a tighter bound of 𝜖: LEMMA 2 (CONVERGENCE OF NON-BACKTRACKING RAN- DOM WALKS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' With the same condition as in Lemma 1, the error 𝜖 by a non-backtracking random walks is less than 𝜖 ≤ ⌈(𝐿 − 1)/3⌉ 𝐿 √︂ log (2/𝛿) 2𝐿𝑀 (4) PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Omitted for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Proof in Supplementary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ■ For example, when using regular random walks of length 𝐿 = 4 with 𝑀 = 30 trials, the estimated error by Lemma 1 with probability 95% is about 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Nevertheless, if we instead use non-backtracking random walks, the error is reduced to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1%, which is 2× lower than the one by regular walks, indicating that the approximated distribution converges well to the true one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In Algorithm 3 line 9, an element-wise multiplication by 𝑨 is done to keep the approximation of 1-hop neighbor for each node, which sufficiently supplies necessary information as well as keeps the resulting matrix sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We use the inverse of the degree matrix 𝑫−1 to reduce the influence of nodes with large de- grees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' This prevents them from dominating the pairwise distance by containing more elements in their rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The element-wise logarithm aims to rescale the distribution in 𝑾, in order to en- large the difference between smaller structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We use SVD for efficient rank-𝑑 decomposition of the sparse proximity matrix 𝑾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We multiply the left-singular vectors 𝑼 by the corresponding squared eigenvalues √ 𝚺 to correct the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Node Similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' To estimate the node similarity, we compute the distance of nodes in the embedding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The intuition is that the nodes that are closer in the embedding space should be better connected with higher-order structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Given the aforementioned embedding 𝑼, the node similarity function S is: S(𝑼𝑖, 𝑼 𝑗) = 𝑒−D(𝑼 𝑖𝑘,𝑼 𝑗𝑘), (5) where 𝑒 ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='718 denotes Euler’s number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Equation 5 is a universal law proposed by Shepard [32], connecting the similarity with distance via an exponential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' While the function D can Under Submission, , Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos be any distance metric, we use Euclidean because it is empirically shown to work well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Negative exponential distribution is used to bound the similarity from 0 to 1, which is close to 0 if the distance is too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Given 𝑨 and 𝑼, “Emphasis” Matrix 𝑨∗ with weighted edges estimated by S is defined in line 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Since S(𝑼𝑖, 𝑼 𝑗) = S(𝑼 𝑗, 𝑼𝑖), 𝑨∗ is still a symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' This is a convenient property, which is later used for the fast computation of the spectral radius (see Lemma 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 Compatibility Matrix Estimation A compatibility matrix contains the class-wise strength of edges and is important for properly inferring the node labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In this subsection, we show how to turn compatibility matrix estimation into an optimization problem by introducing our closed-form formula, which overcomes the defect of edge counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We then illustrate how we conquer several practical challenges to give a precise and fast estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Why NOT Edge Counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The naive way to estimate com- patibility matrix is via counting labeled edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' However, it is inaccurate and has limitations: 1) rare labels will get neglected, and 2) being noisy or biased due to few labeled nodes in real graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The result is even more unreliable if the given labels are imbalanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Figure 3 is an example that edge counting fails if we upsample 10× labels for only class 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' This occurs commonly in practice, since we have only partial labels in node classification tasks, and becomes fatal if the observed distribution is different from the true one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Closed-Form Formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In Equation 1, if we initialize the final belief with the initial one, and omit the addition of the initial belief for the iterative propagation purpose, we have: ˆ𝑩 = 𝑨ˆ𝑬 ˆ𝑯 (6) Our goal is to estimate the compatibility matrix ˆ𝑯 of a given graph, so that the difference between belief propagated by the given priors and the final belief is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' To solve this, we firstly derive the closed-form solution of Equation 6 based on our proposed Network Effect Formula: LEMMA 3 (NETWORK EFFECT FORMULA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Given adja- cency matrix 𝑨 and initial and final beliefs ˆ𝑬 and ˆ𝑩, the closed- form solution of vectorized compatibility matrix vec( ˆ𝑯) is: vec( ˆ𝑯) = (𝑿𝑇 𝑿)−1𝑿𝑇𝒚, (7) where 𝑿 = 𝑰𝑐×𝑐 ⊗ (𝑨ˆ𝑬) and 𝒚 = vec( ˆ𝑩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Omitted for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Proof in Supplementary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ■ Although the final belief matrix ˆ𝑩 is not available before we run actual propagation on the graph, we can replace it by 𝒚 = vec( ˆ𝑬), and extract the ones that are corresponding to the priors P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In other words, we change the problem into minimizing the difference between initial belief of each node 𝑖 ∈ P by the initial beliefs of its neighbors in the priors P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=', 𝑁 (𝑖) ∩ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Intuitively, neighbors should be able to estimate the belief for the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The optimization problem can then be formulated as follows: min ˆ𝑯 ∑︁ 𝑖 ∈P 𝑐∑︁ 𝑢=1 ˆ𝑬𝑖𝑢 − ( 𝑐∑︁ 𝑘=1 ∑︁ 𝑗 ∈𝑁 (𝑖)∩P ˆ𝑬 𝑗𝑘 ˆ𝑯𝑘𝑙) (8) With the help of Network Effect Formula, the optimization prob- lem can then be solved by regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 1 2 3 4 5 6 Class ID 1 2 3 4 5 6 Class ID Edge Counting 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 (a) Balanced Prior 1 2 3 4 5 6 Class ID 1 2 3 4 5 6 Class ID Edge Counting 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 (b) Imbalanced Prior Figure 3: Edge counting can not handle imbalanced case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Class 1 is upsampled in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Data: Emphasis Matrix 𝑨∗, initial belief ˆ𝑬, and priors P Result: Estimated compatibility matrix ˆ𝑯∗ 1 𝒊 ← ∅;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' // indices only related to priors 2 for 𝑝 ∈ P do 3 for 𝑗 = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=',𝑐 do 4 𝒊 ← 𝒊 ∪ {𝑝 + (𝑗 − 1) ∗ 𝑐};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 5 end 6 end 7 𝑿 ← (𝑰𝑐×𝑐 ⊗ (𝑨∗ ˆ𝑬));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' // feature matrix 8 𝒚 ← vec( ˆ𝑬);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' // target vector 9 ˆ𝑯∗ ← 𝑅𝑖𝑑𝑔𝑒𝐶𝑉 (𝑿 [𝒊],𝒚[𝒊]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 10 Return row-normalize(max ( ˆ𝑯∗, 0));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Algorithm 4: Compatibility Matrix Estimation Practical Challenges and Solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Network Effect Formula allows us to estimate the compatibility matrix by solving this op- timization problem, but there still exists two practical challenges that need to be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' First, with few labels, it is difficult to properly separate them into training and validation sets for the regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We thus use ridge regression with leave-one-out cross-validation (RidgeCV) instead of the traditional linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' This allows us to fully exploit the observations without having a bias caused by random splits of training and validation sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Moreover, the regularization effect of ridge regression makes the compatibility matrix more robust to noisy observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' It is noteworthy that the additional computational cost of RidgeCV is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Next, the compatibility matrix estimated with the adjacency matrix 𝑨 is easily interfered with by noisy neighbors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=', weakly- connected pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' To address this issue, we use our proposed “Em- phasis” Matrix 𝑨∗ instead (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1), to pay attention to the labels of neighbors that are structurally important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Since the rows of the estimated matrix 𝑯 do not sum to one in this ap- proach, we filter out the negative values and normalize the sum of each row to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' This is done safely, since the negative values represent negligible relationships between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The overall process of estimation is shown in Al- gorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We extract the indices that are corresponding to the priors after the Kronecker product and vectorization in line 2 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The optimization is then conducted in line 8 to 10 to estimate the compatibility matrix ˆ𝑯∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The negative value filtering and row normalization is done on line 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ULTRAPROP: Principled and Explainable Propagation on Large Graphs Under Submission, , 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3 Theoretical Analysis Convergence Guarantee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' To ensure the convergence of propa- gation, we introduce a scaling factor multiplied to it during the iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The exact convergence of ULTRAPROP is as follows: LEMMA 4 (EXACT CONVERGENCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The criterion for the exact convergence of ULTRAPROP is: ULTRAPROP exactly converges ⇔ 0 < 𝑓 < 1 𝜌(𝑨∗) , (9) where 𝜌(·) denotes the spectral radius of the given matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Omitted for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Proof in Supplementary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ■ A smaller scaling factor leads to a faster convergence, never- theless, distorts the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In ULTRAPROP, we recommend a large eigenvalue close to 1, setting 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='9/𝜌(𝑨∗) as a reason- able default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Since 𝑨∗ is built to be symmetric and sparse (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1), the computation of the spectral radius can be done efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Complexity Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ULTRAPROP uses sparse matrix repre- sentation of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The time complexity is given as: LEMMA 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ULTRAPROP scales linearly on the input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' the time complexity of ULTRAPROP is at most 𝑂(𝑚), (10) and the space complexity is at most 𝑂(max (𝑚,𝑛 · 𝐿 · 𝑀) + 𝑛 · 𝑐2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (11) PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Omitted for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Proof in Supplementary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ■ 5 EXPERIMENTS In this section, we aims to answer the following questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Q1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Accuracy: How well does ULTRAPROP work on real-world graphs as compared to the baselines?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Q2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Scalability: How does the running-time of ULTRAPROP scale w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' graph size?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Explainability: How to explain the results of ULTRAPROP?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Experimental Setup Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We focus on large graphs and include eight graph datasets with at least 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='5K nodes (details in Supplementary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1) in our evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The statistics of datasets are shown in Table 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' For each dataset, we sample only a few node labels as initial beliefs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We do this for five times and report the average and standard deviation to omit the biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' “Synthetic” is the enlarged version of the graph shown in Figure 1, which contains both heterophily and homophily NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Noisy edges are injected in the background, and the dense blocks are constructed by randomly generating higher-order structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We compare ULTRAPROP with five state-of-the-art baselines and separate them into four groups: General GNNs: GCN [16], and APPNP [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Heterophily GNN: MIXHOP [2], and GPR-GNN [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' BP-based methods: HOLS [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Our pro- posed methods: ULTRAPROP-Hom and ULTRAPROP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ULTRA- PROP-Hom is ULTRAPROP using identity matrix as compatibility matrix, which assumes homophily and does not handle NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The details of baselines are given in Supplementary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Experimental Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' For deep graph models, since we fo- cus on the graph without node features, the node degrees are transformed into one hot encoding and used as the node fea- tures, which is suggested and implemented by several studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' GraphSAGE and PyTorch Geometric) [9, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The de- tails of hyperparameters are given in Supplementary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' To give fair comparisons on run time, all the experiments are run on the same machine, which is a stock Linux server with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2GHz Intel Xeon CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2, we further investigate how much the extra cost is, if a more powerful and but more expensive machine is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 Q1 - Accuracy In Table 2 and 3, we report the accuracy and wall-clock time for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We highlight the top three from dark to light by , and denoting the first, second and third place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' OBSERVATION 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ULTRAPROP wins on X-ophily, heterophily and homophily datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' X-ophily and Heterophily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In Table 2, ULTRAPROP outper- forms all the competitors significantly by more than 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4% and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='8% accuracy on the “Synthetic” and “Pokec-Gender” datasets, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' These datasets have strong NE, thus ULTRAPROP boosts the accuracy owing to precise estimations of compatibility matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The success in “Synthetic” further demonstrates its ability to handle the dataset with X-ophily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Heterophily GNNs, namely MIXHOP and GPR-GNN, all fail to predict correctly, giving results close to random guessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' With homophily assumption, General GNNs and BP-based methods also perform poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Both “arXiv-Year” and “Patent-Year” datasets are shown to only have weak NE (in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2), thus resulting in relatively low accuracy for all methods compared with the other two datasets with strong NE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Even so, ULTRAPROP still outperforms the competitors by estimating a reasonable compatibility matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In “arXiv-Year”, ULTRAPROP receives the second place by running 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='6× faster than MIXHOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In “Patent-Year”, only ULTRAPROP, APPNP and MIXHOP are able to give accuracy higher than random guessing, which is 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In the cases that ULTRAPROP is faster than ULTRAPROP- Hom is because of both the low cost of compatibility matrix estimation, and the lower spectral radius of ˆ𝑯∗, leading to a faster convergence while propagating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Homophily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In Table 3, ULTRAPROP-Hom outperforms all the competitors on two homophily datasets, namely “GitHub” and “Pokec-Locality”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ULTRAPROP performs similarly to UL- TRAPROP-Hom, indicating its generalizability to the homophily datasets by estimating near-identity matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In addition, ULTRA- PROP-Hom gives competitive results with HOLS on the other two homophily datasets “Facebook” and “arXiv-Category”, while being 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='9× and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='7× faster than HOLS respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' General GNNs rely heavily on node features for inference which explains their poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' OBSERVATION 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Our optimizations makes difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We evaluate the effect of different compatibility matrices – (i) ULTRAPROP-EC conducts edge counting on the labels of adja- cent nodes in the priors, instead of using our Network Effect Formula, and (ii) ULTRAPROP-A uses the adjacency matrix in- stead of “Emphasis” Matrix to estimate the compatibility matrix Under Submission, , Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos Table 2: ULTRAPROP wins on X-ophily and Heterophily datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Accuracy, running time, and speedup are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Win- ners and runner-ups in , and .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Dataset Synthetic Pokec-Gender arXiv-Year Patent-Year # of Nodes / Edges / Classes 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2M / 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0M / 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='6M / 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3M / 2 169K / 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2M / 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3M / 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3M / 5 Label Fraction 4% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4% 4% 4% NE Strength Strong Strong Weak Weak NE Type X-ophily Heterophily X-ophily Heterophily Method Accuracy (%) Time (s) Speedup Accuracy (%) Time (s) Speedup Accuracy (%) Time (s) Speedup Accuracy (%) Time (s) Speedup GCN 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 3456 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='7× 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 2906 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='9× 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 132 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3× 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 894 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='5 692 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='7× 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3 8139 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='6× ULTRAPROP-Hom 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='7±0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0× 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3 1270 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0× ULTRAPROP 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='5 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0× 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='7 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0× 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='5 122 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0× 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='6±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 1231 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0× Table 4: Ablation Study: Estimating compatibility matrix by the proposed “Emphasis” Matrix is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Accuracy (%) is reported in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Datasets NE Strength ULTRAPROP-Hom ULTRAPROP-EC ULTRAPROP-A ULTRAPROP Synthetic Strong 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='5±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 Pokec-Gender 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='9±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 arXiv-Year (imba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=') Weak 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='5±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='6 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 Patent-Year (imba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=') 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='9 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 Table 5: ULTRAPROP is thrifty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' AWS total dollar amount ($) is reported in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The blue and red fonts denote run- ning a single experiment by t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='small and p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2xlarge, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Accuracy (%) is reported in Table 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Datasets ULTRAPROP GCN Pokec-Gender $ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='28 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0×) $ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='61 (45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0×) Pokec-Locality $ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='47 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0×) $ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='66 (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1×) in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' To demonstrate effectiveness of our proposed estimation over edge counting, we upsample 5% labels to the class with the fewest labels in the datasets with weak NE, which are class 2 in “arXiv-Year” and class 1 in “Patent-Year”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We use the original labels for propagation in the imbalanced datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In Table 4, we find that ULTRAPROP outperforms all its vari- ants in four datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In the datasets with strong NE, ULTRAPROP shows its robustness to the structural noises and gives better re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In the imbalanced datasets, while ULTRAPROP-EC brings its vulnerability to light, ULTRAPROP stays with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' This study highlights the importance of a precise compatibility matrix estimation, as well as forming it into an optimization problem by our Network Effect Formula as shown in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Furthermore, we compare ULTRAPROP with LINBP to dis- play its advantages in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In Figure 4a, the accuracy gap between them indicates the necessity of precisely estimating the compatibility matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In figure 4b, owing to “Emphasis” Matrix, ULTRAPROP-Hom improves the accuracy in all homophily cases 100 101 102 103 Run Time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='8 Accuracy LinBP UltraProp Synthetic Pokec-Gender arXiv-Year Patent-Year Facebook GitHub arXiv-Category Pokec-Locality (a) Run Time vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Accuracy Synthetic Pokec-Gender arXiv-Year Patent-Year Facebook GitHub arXiv-Category Pokec-Locality 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='8 Accruacy LinBP UltraProp-Hom UltraProp 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='8x (b) Accuracy Figure 4: Ablation Study: ULTRAPROP wins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' It provides the best trade-off between accuracy and running time compared with LINBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' compared with LINBP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' owing to both “Emphasis” Matrix and Network Effect Formula, ULTRAPROP improves the accuracy in all cases while adding negligible penalty on run time, provid- ing the best trade-off compared with LINBP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ULTRAPROP per- forming similarly to ULTRAPROP-Hom on homophily datasets, indicates that it correctly estimates near-identity matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ULTRAPROP: Principled and Explainable Propagation on Large Graphs Under Submission, , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 Q2 - Scalability We vary the edge number in “Pokec-Gender” and plot against the wall-clock running time for ULTRAPROP in Figure 1c, in- cluding both training and inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' As there is no good way to sample the graph [19], and also it is prohibitive to use graph generator with million nodes, we try our best to ensure the connectivity by continuously removing the nodes in the graph, until the number of edges is no greater than the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Note that ULTRAPROP scales linearly as expected from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Not only ULTRAPROP is scalable and linear, but it is also thrifty, achieving up to 45× savings in dollar cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' It requires only CPU, while comparable speeds by competitors, require GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Table 5 shows the estimated cost, assuming that we use a small CPU machine for ULTRAPROP, and a GPU machine for GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Details of computation are provided in Supplementary B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3 Q3 - Explainability OBSERVATION 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ULTRAPROP estimated the correct compat- ibility matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We illustrate that the estimations of compatibility matrix by Network Effect Formula are precise in Figure 5, so as to inter- preting the interrelations of classes extremely well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The inter- relations of shown estimated compatibility matrices are similar to the ones of edge counting in Figure 2, while being more ro- bust to the noisy neighbors, namely, weakly connected ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' For “Synthetic”, ULTRAPROP gives the exact answer that we use to generate the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' For “Pokec-Gender”, ULTRAPROP suc- cessfully estimates that people tend to connect to the ones with opposite gender.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' This corresponds to the fact that people incline to have more opposite gender interactions during their reproduc- tive age [11], where the average ages of male and female in the dataset are 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4 and 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Although “arXiv-Year” and “Patent-Year” do not have strong NE, ULTRAPROP still gives an estimated compatibility matrices making much sense in the real world, where the papers and patents only cite to the ones whose published dates are relatively close to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We omit the re- sults on homophily datasets, for brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In all cases ULTRAPROP resulted in an near-identity compatibility matrix, as expected, supported by giving similar results as ULTRAPROP-Hom, which uses identity matrix as compatibility matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 6 CONCLUSIONS We firstly presented Network Effect Analysis (NEA) to identify whether a graph exhibit network-effect or not, and surprisingly dis- cover the absence of it in many real-world graphs known to have heterophily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Next, we present ULTRAPROP to solve node classi- fication based two insights, network-effect (NE) and neighbor- differentiation (ND), which has the following advantages: (1) Accurate: thanks to the precise compatibility matrix esti- mation by NE, and ND that weighs important neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (2) Explainable: it interprets interrelations of classes with the estimated compatibility matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (3) Scalable: it scales linearly with the input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (4) Principled: it provides provable guarantees (Lemma 1, 2 and 4) and closed-form solution (Lemma 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Applied on real-world million-scale graph datasets with over 22M edges, ULTRAPROP only requires 12 minutes on a stock 1 2 3 4 5 6 Class ID 1 2 3 4 5 6 Class ID Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Matrix 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 (a) “Synthetic”: X-ophily with Strong NE 1 2 Class ID 1 2 Class ID Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Matrix 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0 (b) “Pokec-Gender”: Heterophily with Strong NE 1 2 3 4 5 Class ID 1 2 3 4 5 Class ID Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Matrix 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='8 (c) “arXiv-Year”: X-ophily with Weak NE 1 2 3 4 5 Class ID 1 2 3 4 5 Class ID Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Matrix 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='6 (d) “Patent-Year”: Heterophily with Weak NE Figure 5: ULTRAPROP is explainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The estimated com- patibility matrices are similar to the edge counting matrix (in Figure 2), while being robust to the noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' CPU-machine, and outperforms recent baselines on accuracy, as well as on speed (≥ 9×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Reproducibility: Our implemented source code and prepro- cessed datasets will be published once the paper is accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Under Submission, , Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos REFERENCES [1] Nvidia rtx a6000 deep learning benchmarks.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Heimann, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Akoglu, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Koutra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Beyond homophily in graph neural networks: Current limitations and effective designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 33:7793–7804, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ULTRAPROP: Principled and Explainable Propagation on Large Graphs Under Submission, , A PROOF A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 Proof of Lemma 1 and 2 PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' For a 𝐿-steps random walk sequence 𝑆 with 𝑀 trials, the sequence length |𝑆| is 𝐿𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We define the random variable 𝑋, denoting the probability of node 𝑖 will walk to its 𝑗-th neighbor: 𝑋 = P(node 𝑖 walks to 𝑁 (𝑖)𝑗) = �|𝑆 | 𝑘=1 1(𝑁 (𝑖)𝑗 = 𝑆𝑘) |𝑆| , (12) where P denotes the probability and 1 denotes the indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' With regular random walk in the graph without self-loops, the random variable 𝑋 is upper-bounded by ⌈(𝐿−1)/2⌉ 𝐿 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We can thus apply Hoeffding’s inequality: P(| ˆ𝜇|𝑆 | − 𝜇| ≥ 𝜖) ≤ 2 exp −2𝐿3𝑀𝑡2 ⌈(𝐿 − 1)/2⌉2 , (13) where ˆ𝜇|𝑆 | denotes the sampled mean of the given random vari- able, and 𝜇 denotes the expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Let 𝛿 = 2 exp −2𝐿3𝑀𝑡2 ⌈(𝐿−1)/2⌉2 , with probability 1 − 𝛿, the error 𝜖 is: 𝜖 = | ˆ𝜇|𝑆 | − 𝜇| ≤ ⌈(𝐿 − 1)/2⌉ 𝐿 √︂ log (2/𝛿) 2𝐿𝑀 (14) With the help of non-backtracking random walk [3], we can further shrink the upper bound of 𝑋 into ⌈(𝐿−1)/3⌉ 𝐿 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Now, let 𝛿 = 2 exp −2𝐿3𝑀𝑡2 ⌈(𝐿−1)/3⌉2 , with probability 1 − 𝛿, the error 𝜖 can thus be improved to: 𝜖 = | ˆ𝜇|𝑆 | − 𝜇| ≤ ⌈(𝐿 − 1)/3⌉ 𝐿 √︂ log (2/𝛿) 2𝐿𝑀 (15) ■ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 Proof of Lemma 3 PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In the beginning, we introduce two necessary nota- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' vec(·) denotes the vectorization operator: vec(𝑿) = [𝑿11, · · · , 𝑿𝑚1, 𝑿12, · · · , 𝑿𝑚2, · · · , 𝑿𝑚𝑛]⊤, (16) where 𝑿 is an 𝑚 ×𝑛 matrix, and 𝑿𝑖𝑗 denotes the element of 𝑿 on the 𝑖-th row and the 𝑗-th column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Next, the Knronecker product of given two 𝑚 × 𝑛 matrices 𝑿 and 𝒀 is: 𝑿 ⊗ 𝒀 = \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 𝑿11𝒀 𝑿12𝒀 · · 𝑿1𝑛𝒀 𝑿21𝒀 𝑿22𝒀 · · 𝑿2𝑛𝒀 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 𝑿𝑚1𝒀 𝑿𝑚2𝒀 · · 𝑿𝑚𝑛𝒀 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb (17) The idea of this proof is to reformulate the equation in order to derive the final result by the closed formula of Linear Regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We firstly show two well-known equations that will be used in our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Given the features 𝑿 and target 𝒚, the closed formula of the weights 𝑾 of Linear Regression is: 𝑾 = (𝑿𝑇 𝑿)−1𝑿𝑇𝒚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' (18) The famous property of the mixed Kronecker matrix-vector prod- uct [4] is also used: vec(𝑩𝑽𝑨𝑇 ) = (𝑨 ⊗ 𝑩)𝒗, (19) where the matrix 𝑽 = vec−1(𝒗) is the result of the inverse of the vectorization operator on 𝒗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' To begin the derivation, we vectorize Equation 6 into: vec( ˆ𝑩) = vec((𝑨ˆ𝑬) ˆ𝑯𝑰𝑐×𝑐), (20) where 𝑰𝑐×𝑐 is a 𝑐 × 𝑐 identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The trick here, which is the key of this proof, is to multiply one more identity matrix by ˆ𝑯.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Therefore, we use Equation 19 to reformulate the equation to: vec( ˆ𝑩) = (𝑰𝑐×𝑐 ⊗ (𝑨ˆ𝑬))vec( ˆ𝑯) (21) By letting 𝑿 = 𝑰𝑐×𝑐 ⊗ (𝑨ˆ𝑬) and 𝒚 = vec( ˆ𝑩) in Equation 18, we can then derive the closed-form solution of vectorized compati- bility matrix as follows: vec( ˆ𝑯) = (𝑿𝑇 𝑿)−1𝑿𝑇𝒚 (22) ■ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3 Proof of Lemma 4 PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ULTRAPROP exactly converges if and only if 𝜌(𝑨∗)𝜌( ˆ𝑯∗) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' However, the compatibility matrix 𝑯∗ is row- normalized, so the largest eigenvalue 𝜌(𝑯∗) = 1 is a constant, and is less than one after centering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Thus, the scaling factor 𝑓 multiplied to the propagation (in Algorithm 2 line 5) should be in the range of (0, 1 𝜌 (𝑨∗) ) to meet the criterion of exact convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ■ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4 Proof of Lemma 5 PROOF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In the neighbor-differentiation phase, for each ran- dom walk, each node visits at most 𝐿·𝑀 unique nodes, so the max- imum number of non-zero elements in 𝑾 is either 𝑛 · 𝐿 · 𝑀 if we have not walked through all the edges, or 𝑚 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The time complexity of SVD on 𝑾 then takes 𝑂(𝑑 · max (𝑚,𝑛 · 𝐿 · 𝑀)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In the network-effect phase, the time complexity for the Fisher’s ex- act test is 𝑂(max (𝑪)), where max (𝑪) is a constant bounded by 500 in our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Therefore, network-effect analysis takes 𝑂(|𝒆 ′| · 𝑐2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' For the regression, since there are 𝑐 sets of pa- rameters are independent, we can separate the problem into 𝑐 tasks, where each contains 𝑐 features and |𝒑| samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Thus the complexity can be reduced to 𝑂(|𝒑| · 𝑐3), and the efficient leave-one-out cross-validation only needs to be done once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' In the propagation phase, it takes at most 𝑂(𝑚 + 𝑛) for sparse matrix multiplication to run 𝑡 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Thus, the time complexity is 𝑂(𝑑 max (𝑚,𝑛 · 𝐿 · 𝑀) + |𝒑| ·𝑐3 +𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' However, in practice, 𝑐, |𝒑| and 𝑡 are usually small constants which are negligible, and 𝑚 is usually much larger than them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Therefore, keeping only the dominating terms, the time complexity is approximately 𝑂(𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 𝑾 contains at most max (𝑚,𝑛 · 𝐿 · 𝑀) non-zero elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The Kronecker product at most contains 𝑛 · 𝑐2 non-zero elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ˆ𝑩 and ˆ𝑯 contain at most 𝑛 ·𝑐 and 𝑐2 non-zero elements, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Thus, the space complexity is 𝑂(max (𝑚,𝑛 · 𝐿 · 𝑀) + 𝑛 · 𝑐2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' ■ B REPRODUCIBILITY B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 Datasets “Pokec-Gender” [33] is an online social network in Slo- vakia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' [23] re-labels the nodes by users’ genders instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' “arXiv-Year” [14] is a citation network between all Com- puter Science arXiv papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' [23] re-labels the nodes by the posted years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' “Patent-Year” [20] is the patent citation network from 1980 to 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' [23] re-labels the nodes by the application year, bucketized into five consecutive 3-year ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' “Synthetic” is a graph enlarged by the one in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' It contains both heterophily and homophily network-effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Under Submission, , Meng-Chieh Lee, Shubhranshu Shekhar, Jaemin Yoo, and Christos Faloutsos Table 6: Hyperparameters for Deep Graph Models Method Hyperparameters GCN lr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='01, wd=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0005, hidden=16, dropout=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='5 APPNP lr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='002, wd=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0005, hidden=64, dropout=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='5, K=10, alpha=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 MIXHOP lr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='01, wd=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0005, cutoff=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1, layers1=[200, 200, 200], layers2=[200, 200, 200] GPR-GNN lr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='002, wd=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='0005, hidden=64, dropout=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='5, K=10, alpha=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='1 Noisy edges are randomly injected in the background, and the dense blocks are constructed by randomly creating higher-order structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' “Facebook” [30] is a page-to-page network of verified Facebook sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Nodes are labeled by the categories such as politicians and companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' “GitHub” [30] is a social network of developers in June 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Nodes are labeled as web or a machine learning developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' “arXiv-Category” [37] is the same dataset as the arXiv- Year dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Nodes are labeled by the primary categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' “Pokec-Locality” [33] is the same dataset as the Pokec- Gender dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Nodes are labeled by the uses’ localities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2 Baselines GCN2 [16] is a well-known deep graph model, learning and aggregating the weights of two-hop neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' APPNP4 [17] utilizes personalized PageRank to leverage the local information and a larger neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' MIXHOP3 [2] mixes powers of the adjacency matrix to incorporate more than 1-hop neighbors in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' GPR-GNN4 [6] allows the learnable weights to be nega- tive during propagation with Generalized PageRank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' HOLS5 [8] is a label propagation method with attention, by increasing the importance of a neighbor if they appear in the same motif at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3 Hyperparameters For ULTRAPROP and ULTRAPROP-Hom, we use random walks of length 4 with 10 trials except GitHub, arXiv-Category and Pokec-Locality datasets, where we use 30 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The decompo- sition rank is set to be 128, which is empirically shown to be enough in the embedding tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The weights of HOLS for differ- ent motifs are set to be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' For the deep graph models, under the setting that the given labels are very few, it is impossible to separate a validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' We then train them for a fixed number of epochs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 200 epochs), which is usually sufficient enough for them to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' All the fully connected layers are replaced by the sparse version in order to fit into memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Both adjacency ma- trices and features are normalized and turn into sparse matrices if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' For other hyperparameters, we use the default settings given by the authors, and give the details in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='com/tkipf/pygcn 3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='com/benedekrozemberczki/MixHop-and-N-GCN 4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='com/jianhao2016/GPRGNN 5https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='com/dhivyaeswaran/hols B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='4 Scalability We select machines provided by AWS with comparable specs as we use for the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' For CPU machine, we select t3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='small with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='3GHz CPU and 2GB RAM, which is faster than ours, and costs $0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='023 per hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' For GPU machine, we select p3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='2xlarge with a V100 GPU, which costs $3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='06 per hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' According to [1], it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content='89 slower than the RTX A6000 GPU we use on running PyTorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' The running time of GCN on “Pokec-Gender” and “Pokec-Locality” are 673 and 730 seconds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} +page_content=' Using the provided information, the results in Table 5 can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2NAyT4oBgHgl3EQfbve8/content/2301.00270v1.pdf'} diff --git a/3dE2T4oBgHgl3EQf6Ago/content/tmp_files/2301.04195v1.pdf.txt b/3dE2T4oBgHgl3EQf6Ago/content/tmp_files/2301.04195v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..364863cf3eb782c25cc9595dd24ba05129aa0ade --- /dev/null +++ b/3dE2T4oBgHgl3EQf6Ago/content/tmp_files/2301.04195v1.pdf.txt @@ -0,0 +1,1198 @@ +ORBIT: A Unified Simulation Framework for +Interactive Robot Learning Environments +Mayank Mittal1,2, Calvin Yu3, Qinxi Yu3, Jingzhou Liu1,3, Nikita Rudin1,2, David Hoeller1,2, +Jia Lin Yuan3, Pooria Poorsarvi Tehrani3, Ritvik Singh1,3, Yunrong Guo1, Hammad Mazhar1, +Ajay Mandlekar1, Buck Babich1, Gavriel State1, Marco Hutter2, Animesh Garg1,3 +Fig. 1: ORBIT framework provides a large set of robots, sensors, rigid and deformable objects, motion generators, and teleoperation +interfaces. Through these, we aim to simplify the process of defining new and complex environments, thereby providing a common +platform for algorithmic research in robotics and robot learning. +Abstract— We present ORBIT, a unified and modular frame- +work for robot learning powered by NVIDIA Isaac Sim. It +offers a modular design to easily and efficiently create robotic +environments with photo-realistic scenes and fast and accurate +rigid and deformable body simulation. With ORBIT, we provide +a suite of benchmark tasks of varying difficulty– from single- +stage cabinet opening and cloth folding to multi-stage tasks +such as room reorganization. To support working with diverse +observations and action spaces, we include fixed-arm and +mobile manipulators with different physically-based sensors +and motion generators. ORBIT allows training reinforcement +learning policies and collecting large demonstration datasets +from hand-crafted or expert solutions in a matter of minutes +by leveraging GPU-based parallelization. In summary, we offer +an open-sourced framework that readily comes with 16 robotic +platforms, 4 sensor modalities, 10 motion generators, more than +20 benchmark tasks, and wrappers to 4 learning libraries. With +this framework, we aim to support various research areas, +including representation learning, reinforcement learning, imi- +tation learning, and task and motion planning. We hope it helps +establish interdisciplinary collaborations in these communities, +and its modularity makes it easily extensible for more tasks +and applications in the future. For videos, documentation, and +code: https://isaac-orbit.github.io/. +I. INTRODUCTION +The recent surge in machine learning has led to a paradigm +shift in robotics research. Methods such as reinforcement +learning (RL) have shown incredible success in challenging +problems such as quadrupedal locomotion [1], [2], [3] and +in-hand manipulation [4], [5]. However, learning techniques +1 NVIDIA, 2 ETH Z¨urich, 3 University of Toronto, Vector Institute. +Correspondence: mittalma@ethz.ch, garg@cs.toronto.edu. +require a wealth of training data, which is often challenging +and expensive to obtain at scale on a physical system. This +makes simulators an appealing alternative for developing +systems safely, efficiently, and cost-effectively. +An ideal robot simulation framework needs to provide fast +and accurate physics, high-fidelity sensor simulation, diverse +asset handling, and easy-to-use interfaces for integrating new +tasks and environments. However, existing frameworks often +make a trade-off between these aspects depending on their +target application. For instance, simulators designed mainly +for vision, such as Habitat [18] or ManipulaTHOR [16], offer +decent rendering but simplify low-level interaction intricacies +such as grasping. On the other hand, physics simulators for +robotics, such as IsaacGym [17] or Sapien [15], provide fast +and reasonably accurate rigid-body contact dynamics but do +not include physically-based rendering (PBR), deformable +objects simulation or ROS [19] support out-of-the-box. +In this work, we present ORBIT an open-source frame- +work, built on NVIDIA Isaac Sim [20], for intuitive designing +of environments and tasks for robot learning with photo- +realistic scenes and state-of-the-art physics simulation. Its +modular design supports various robotic applications, such as +reinforcement learning (RL), learning from demonstrations +(LfD), and motion planning. Through careful design of inter- +faces, we aim to support learning for a diverse range of robots +and tasks, allowing operation at different levels of observa- +tion (proprioception, images, pointclouds) and action spaces +(joint space, task space). To ensure high-simulation through- +put, we leverage hardware-accelerated robot simulation, and +include GPU implementations for motion generation and +arXiv:2301.04195v1 [cs.RO] 10 Jan 2023 + +TABLE I: Comparison between different simulation frameworks and ORBIT. The check (✓) and cross (X) denote presence or absence of +the feature. In Robotic Platforms column, M stands for manipulator. In Scene Authoring column, G stands for game-based designing, +M for mesh-scan scenes, and P for procedural-generation. +Vectorization +Supported Dynamics +Sensors +Robotic Platforms +Name +Physics Engine +Renderer +CPU +GPU +Rigid +Cloth +Soft +Fluid +PBR +Tracing +RGBD +Semantic +LiDAR +Contact +Fixed-M +Mobile-M +Legged +Scene +Authoring +MetaWorld [6] +MuJoCo +OpenGL +✓ +X +✓ +X +X +X +X +X +X +X +X +✓ +X +X +P +RoboSuite [7] +MuJoCo +OpenGL, OptiX +✓ +X +✓ +X +X +X +X +✓ +✓ +X +✓ +✓ +X +X +P +DoorGym [8] +MuJoCo +Unity +X +X +✓ +X +X +X +✓ +✓ +✓ +X +X +✓ +X +X +P, G +DEDO [9] +Bullet +OpenGL +✓ +X +✓ +✓ +✓ +X +X +✓ +X +X +X +✓ +X +X +P, G +RLBench [10] +Bullet/ODE +OpenGL +X +X +✓ +X +X +X +X +✓ +✓ +X +✓ +✓ +X +X +P +iGibson [11] +Bullet +MeshRenderer +✓ +X +✓ +X +X +X +✓ +✓ +✓ +✓ +X +X +✓ +X +M +Habitat 2.0 [12] +Bullet +Magnum +X +X +✓ +X +X +X +✓ +✓ +✓ +X +X +✓ +✓ +✓ +P, M +SoftGym [13] +FleX +OpenGL +✓ +X +✓ +✓ +✓ +✓ +X +X +X +X +X +✓ +X +X +P +ThreeDWorld [14] +PhysX 4/FleX/Obi +Unity3D +X +X +✓∗ +✓∗ +✓∗ +✓∗ +✓ +✓ +✓ +X +X +X +✓ +X +P +SAPIEN [15] +PhysX 4 +OptiX, Kuafu +✓ +X +✓ +X +X +X +✓ +✓ +✓ +X +✓ +✓ +✓ +X +P +ManipulatorThor [16] +PhysX 4 +Unity +X +X +✓ +X +X +X +✓ +✓ +✓ +X +X +X +✓ +X +P, G +IsaacGymEnvs [17] +PhysX 5 +Vulkan +✓ +✓ +✓ +X +X +X +X +✓ +✓ +X +✓ +✓ +X +✓ +P +ORBIT (ours) +PhysX 5 +Omniverse RTX +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +✓ +P, M, G +* ThreeDWorld supports simulation of rigid bodies and deformable bodies based on whether PhysX 4 or FleX/Obi is enabled respectively. Thus, it is limited in simulating interactions between rigid and deformable bodies. +observations processing. This allows training and evaluation +of a complete robotic system at scale, without abstracting +out low-level details in robot-environment interactions. +The release of ORBIT v1.0 features: +1) models for three quadrupeds, seven robotic arms, four +grippers, two hands, and four mobile manipulators; +2) a selection of CPU and GPU-based motion generators +implementations for each robot category, including pre- +trained locomotion policies, inverse kinematics, opera- +tional space control, and model predictive control; +3) utilities for collecting human demonstrations using pe- +ripherals (keyboard, gamepad or 3D mouse), replaying +demonstration datasets, and utilizing them for learning; +4) a suite of standardized tasks of varying complexity for +benchmark purposes. These include eleven rigid object +manipulation, thirteen deformable object manipulation, +and two locomotion environments. Within each task, we +allow switching robots, objects, and sensors easily. +In the remaining of the paper, we describe the underlying +simulation choices (Sec. II), the framework’s design deci- +sions and abstractions (Sec. III), and its highlighted features +(Sec. IV). We demonstrate the framework’s applicability for +different workflows (Sec. V) – particularly RL using various +libraries, LfD with robomimic [21], motion planning [22], +[23], and connection to physical robots for deployment. +II. RELATED WORK +Recent years have seen several simulation frameworks, +each specializing for particular robotic applications. In this +section, we highlight the design choices crucial for building +a unified simulation platform and how ORBIT compares to +other frameworks (also summarized in Table I). +a) Physics Engine: Increasing the complexity and re- +alism of physically simulated environments is essential for +advancing robotics research. This includes improving the +contact dynamics, having better collision handling for non- +convex geometries (such as threads), stable solvers for de- +formable bodies, and high simulation throughput. +Prior frameworks [7], [10] using MuJoCo [24] or Bul- +let [25] focus mainly on rigid object manipulation tasks. +Since their underlying physics engines are CPU-based, they +need CPU clusters to achieve massive parallelization [17]. On +the other hand, frameworks for deformable bodies [9], [13] +mainly employ Bullet [25] or FleX [26], which use particle- +based dynamics for soft bodies and cloth simulation. How- +ever, limited tooling exists in these frameworks compared +to those for rigid object tasks. ORBIT aims to bridge this +gap by providing a robotics framework that supports rigid +and deformable body simulation via PhysX SDK 5 [27]. In +contrast to other engines, it features GPU-based hardware +acceleration for high throughput, signed-distance field (SDF) +collision checking [28], and more stable solvers based on +finite elements for deformable body simulation. +b) Sensor simulation: Various existing frameworks [7], +[10], [17] use classic rasterization that limits the photo- +realism in the generated images. Recent techniques [29], [30] +simulate the interaction of rays with object’s textures in a +physically correct manner. These methods helps capture fine +visual properties such as transparency and reflection, thereby +are promising for bridging sim-to-real visual domain gap. +While recent frameworks [15], [12], [16] include physically- +based renderers, they mainly support camera-based sen- +sors (RGB, depth). This is insufficient for certain mobile +robot applications that need range sensors, such as LiDARs. +Leveraging the ray-tracing technology in NVIDIA Isaac Sim, +ORBIT supports all these modalities and includes APIs to +obtain additional information such as semantic annotations. +c) Scene designing and asset handling: Frameworks +support scene creation procedurally [6], [7], [15], via mesh +scans [11], [12] or through game-engine style interfaces [31], +[14]. While mesh scans simplify generating large amounts of +scenes, they often suffer from geometric artifacts and lighting +problems. On the other hand, procedural generation allows +leveraging object datasets for diverse scenes. To not restrict +to either possibility, we facilitate scene designing by using +graphical interfaces and also providing tools for importing +different datasets [32], [33], [34]. +Simulators are typically general-purpose and expose ac- +cess to various internal properties, often alienating non- +expert users due to a steep learning curve. ORBIT inherits +many utilities from the NVIDIA Omniverse and Isaac Sim +platforms, such as high-quality rendering, multi-format asset +import, ROS support, and domain randomization (DR) tools. +However, its contributions lie in the specialization of inter- +faces for robot learning that simplify environment designing +and facilitate transfer to a real robot. For instance, we provide +unified abstractions for different robot and object types, allow + +Fig. 3: ORBIT’s abstractions comprise World, analogous to the real world, and Agent, the computation graph behind the embodied system. +The nodes in the agent’s graph can perform observation-based or action-based processing. Through a graph-cut over this computation +graph and specifying an extrinsic goal, it is feasible to design different tasks within the same World instance. +injecting actuator models into the simulation to assist in +sim-to-real transfer, and support various peripherals for data +collection. Overall, it provides a highly featured state-of-the- +art simulation framework (Table I) while preserving usability +through intuitive abstractions. +III. ORBIT: ABSTRACTIONS AND INTERFACES DESIGN +At a high level, the framework design comprises a world +and an agent, similar to the real world and the software +stack running on the robot. The agent receives raw observa- +tions from the world and computes the actions to apply on the +embodiment (robot). Typically in learning, it is assumed +that all the perception and motion generation occurs at the +same frequency. However, in the real world, that is rarely the +case: (1) different sensors tick at differing frequencies, (2) +depending on the control architecture, actions are applied at +different time-scales [35], and (3) various unmodeled sources +cause delays and noises in the real system. In ORBIT, we +carefully design the interfaces and abstractions to support +(1) and (2), and for (3), we include implementation of +different actuator and noise models as part of the robot +and sensors respectively. +a) World: Analogous to the real world, we define a +world where robots, sensors, and objects (static +or dynamic) exist on the same stage. The world can be de- +signed procedurally (script-based), via scanned meshes [33], +[32], through the game-based GUI of Isaac Sim, or a +combination of them, such as importing scanned meshes +and adding objects to it. This flexibility reaps the benefits +of 3D reconstructed meshes, which capture various archi- +tectural layouts, with game-based designing, that simplifies +the experience of creating and verifying the scene physics +properties by playing the simulation. +Robots are a crucial component of the world since +they serve as the embodiment for interaction. They consist +of an articulated system, sensors, and low-level controllers. +The robot class loads its model from USD files. It may +DC Motor + Actuator Net +(MLP/LSTM) +Fig. 4: Illustration of actuator groups for a legged mobile manipu- +lator. This allows decomposing a complex system into sub-groups +and defining specific transmission models for each of them flexibly. +have onboard sensors specified through the same USD file or +configuration files. The low-level controller processes input +actions through the configured actuator models and applies +desired joint position, velocity, or torque commands to the +simulator (as shown in Fig. 4). The actuator dynamics can be +modeled using first-principle from physics or be learned as +neural networks. This allows injection of real world actuator +characteristics into simulation thereby facilitating sim-to-real +transfer of control policies [36]. +Sensors may exist both on the articulation (as part of the +robot) or externally (such as, third-person cameras). ORBIT +interface unifies different physics-based (range, force, and +contact sensor) and rendering-based (RGB, depth, normals) +sensors under a common interface. To simulate asynchronous +sensing and actuation, each sensor has an internal timer that +governs its operating frequency. The sensor only reads the +simulator buffers at the configured frequency. Between the +timesteps, the sensor returns the previously obtained values. +Objects are passive entities in the world. While several +objects may exist in the scene, the user can define objects +of interest for a specified task and retrieve data/properties +only for them. Object properties mainly comprise visual and +collision meshes, textures, and physics materials. For any +given object, we support randomization of its textures and +physics properties, such as friction and joint parameters. + +rt +Learning +Learning +Task Logic +Task +Agent +Rewards/Costs +Oracle Reset +World +Agent +Sensors +Ot +Node 1 +Passive +Camera +External Sensors +Objects +Node 2 +LiDAR +Robot +at +.... +Node n +Actuator Model +Height Scan +Visualization +On-board +Sensors +Markers +Contact Report +Computation Nodes +O NVIDIA Isaac Sim +Motion Generation +Perception +Filtering +Learning-based +PhysX +NVIDIA +可 +USD +Mapping +Model-based +Iray +by NVIDIAGripper +open/close +joint velocity +Mimic Group +(1) +(6) +Actions +joint position +Arm +joint torque +DC Motor +(6) +(6) +Base +joint position +Actuator Net +joint torque +(12) +(MLP/LSTM) +(12)Rigid +Articulated +Deformable +Cloth +IK +OSC +RMPFlow +OCS2 +NN Policy +Teleoperation +End-Effector +Arm +Mobile Base +Height Scan +Camera +Contact Reporter +Proprioception +Fig. 5: Overview of features included in ORBIT. We provide models of different sensors, robotic platforms, objects from different +datasets, motion generators and teleoperation devices. Using RTX-accelerated ray-tracing, we can obtain high-fidelity images in real-time +for different modalities such as RGB, depth, surface normal, instance and semantic segmentation (pixel-wise and bounding boxes). +b) Agent: An agent refers to the decision-making +process (“intelligence”) guiding the embodied system. While +roboticists have embraced the modularity of ROS [19], most +robot learning frameworks often focus only on the environ- +ment definition. This practice requires code replication, & +adds friction to switching between different implementations. +Keeping modularity at its core, an agent in ORBIT com- +prises various nodes that formulate a computation graph +exchanging information between them. Broadly, we consider +nodes are of two types: 1) perception-based i.e., they process +inputs into another representation (such as RGB-D image to +point-cloud/TSDF), or 2) action-based i.e., they process in- +puts into action commands (such as task-level commands to +joint commands). Currently, the flow of information between +nodes happens synchronously via Python, which avoids the +data exchange overhead of service-client protocols. +c) Learning task and agent: Paradigms such as RL +require specification of a task, a world and may include +some computation nodes of the agent. The task logic +helps specify the goal for the agent, compute metrics (re- +wards/costs) to evaluate the agent’s performance, and manage +the episodic resets. With this component as a separate mod- +ule, it becomes feasible to use the same world definition +for different tasks, similar to learning in the real world, +where tasks are specified through extrinsic reward signals. +The task definition may also contain different nodes of the +agent. An intuitive way to formalize this is by considering +that learning for a particular node happens through a graph +cut on the agent’s computation graph. +To further concretize the design motivation, consider the +example of learning over task space instead of low-level joint +actions for lifting a cube [35]. In this case, the task-space +controller, such as inverse kinematics (IK), would typically +run at 50Hz, while the joint controller requires commands +at 1000 Hz. Although the task-space controller is a part of +the agent’s and not the world’s computation, it is possible +to encapsulate that into the task design. This functionality +easily allows switching between motion generators, such as +IK, operational-space control (OSC), or reactive planners. +IV. ORBIT: FEATURES +While various robotic benchmarks have been proposed [9], +[6], [10], the right choice of necessary and sufficient tasks to +demonstrate “intelligent” behaviors remains an open ques- +tion. Instead of being prescriptive about tasks, we provide +ORBIT as a platform to easily design new tasks. To facilitate +the same, we include a diverse set of supported robots, +peripheral devices, and motion generators and a large set +of tasks for rigid and soft object manipulation for essential +skills such as folding cloth, opening the dishwasher, and +screwing a nut into a bolt. Each task showcases aspects of +physics and renderer that we believe will facilitate answering +crucial research questions, such as building representations +for deformable object manipulation and learning skills that +generalize to different objects and robots. +a) Robots: We support 4 mobile platforms (one om- +nidirectional drive base and three quadrupeds), 7 robotic +arms (two 6-DoF and five 7-DoF), and 6 end-effectors (four +parallel-jaw grippers and two robotic hands). We provide +tools to compose different combinations of these articulations +into a complex robotic system such as a legged mobile +manipulator. This provides a large set of robot platforms, +each of which can be switched in the World. +b) I/O Devices: Devices define the interface to periph- +eral controllers that teleoperate the robot in real-time. The +interface reads the input commands from an I/O device and +parses them into control commands for subsequent nodes. +This helps not only in collecting demonstrations [21] but also +in debugging the task designs. Currently, we include support +for Keyboard, Gamepad (Xbox controller), and Spacemouse +from 3Dconnexion. +c) Motion Generators: Motion generators transform +high-level actions into lower-level commands by treating +input actions as reference tracking signals. For instance, +inverse kinematics (IK) [37] interprets commands as the +desired end-effector poses and computes the desired joint +positions. Employing these controllers, particularly in task +space, has shown to help sim-to-real transferability of robot +manipulation policies [7], [35]. +With ORBIT, we include GPU-based implementations for: +differential IK [37], operational-space control [38] and joint- +level control. Additionally, we provide CPU implementa- +tion of state-of-the-art model-based planners such as RMP- +Flow [22] for fixed-arm manipulators and OCS2 [23] for +whole-body control of mobile manipulators. We also provide +pre-trained policies for legged locomotion [39] to facilitate +solving navigation tasks using base velocity commands. + +40%MORE +French's +YELLOL3DconnexionF1 +F4 +F6 +F7 +F8 +SYGR +Lock +Bresk +2 +3 +6 +7 +8 +Q +R +T +Home +Pgup +Cops Lock +G +H +Enter +Doier +Booe +Z +X +tsift +B +tshint +Pon +Ente +Alt +Alt +Ctrt11GPU +IK +OSC +NN PolicyCPU +OCS2 +RMPF1owated +Fluid +ClotlTeleoperationFig. 6: Demonstration of the designed tasks using hand-crafted state machines and task-space controllers. Leveraging recent advances +in physics engines, we support high-fidelity simulation of rigid and deformable objects. We include environments that allow switching +between robots, objects, observations, and action spaces through configuration files (Task videos). +d) Rigid-body Environments: For rigid-body environ- +ments, it is vital to have accurate contact physics, fast +collision checking, and articulate joints simulation. While +some of these tasks exist in prior works [6], [10], [28], +[39], we enhance them with our framework’s interfaces and +provide more variability using DR tools. We also extend ma- +nipulation tasks for fixed-arm robots to mobile manipulators. +For brevity, we list the environments are as follows: +1) Reach - Track desired pose of the end-effector. +2) Lift - Take an object to a desired position. +3) Beat the Buzz - Displace a key around a pole +without touching the pole. +4) Nut-Bolt - Tighten a nut on a given bolt. +5) Cabinet - Open or close a cabinet (articulated object). +6) Pyramid Stack - Stack blocks into pyramids. +7) Hockey [10] - Shoot a puck into the net using a stick. +8) Peg In Hole - Insert blocks into their holes. +9) Jenga [10] - Remove and stack blocks into a tower. +10) In-Hand Repose - Using dexterous robotic hands. +11) Velocity Locomotion - Track a desired velocity +command via a legged robot on various terrains. +e) Deformable-body Environments: +Deformable ob- +jects have a high dimensional state and complex dynamics +which are difficult to capture succinctly for robot learning. +With ORBIT, we provide seventeen deformable objects assets +(such as toys and garments) with valid physics configurations +and methods to generate new assets (such as rectangular +cloth) procedurally. A concise list of included environments +are as follows: +1) Cloth Lifting - Lift a cloth to a target position. +2) Cloth Folding - Fold a cloth into a desired state. +3) Cloth Spreading - Spread a cloth on a table. +4) Cloth Dropping - Drop a cloth into a container. +5) Flag Hoisting - Hoist a flag standing on a table. +6) Soft Lifting - Lift a soft object to a target position. +7) Soft Placing - Place a soft object on a shelf. +8) Soft Stacking - Stack soft objects on each other. +9) Soft Dropping - Drop soft objects into a container. +10) Tower of Hanoi - Stack toruses around a pole. +11) Rope Reshaping - Reshape a rope on a table. +12) Fluid Pouring - Pour fluid into another container. +13) Fluid Transport - Move a filled container without +causing any spillages. +It should be noted that the environments (1), (2), and +(3) carry the same World definition. They only differ in +their task logic module, i.e. the associated reward associ- +ated, which is defined through configuration managers. This +modularity allows code reusage and makes it easier to define +a large set of tasks within the same World. +V. EXEMPLAR WORKFLOWS WITH ORBIT +ORBIT is a unified simulation infrastructure that provides +both pre-built environments and easy-to-use interfaces that +enables extendability and customization. Owing to high- +quality physics, sensor simulation, and rendering, ORBIT us +useful for multiple robotics challenges in both perception +and decision-making. We outline a subset of such use cases +through exemplar workflows. +A. GPU-based Reinforcement Learning +We provide wrappers to different RL frameworks (rl- +games [40], RSL-rl [39], and stable-baselines-3 [41]). This +allows users to test their environments on a larger set of RL +algorithms and facilitate algorithmic developments in RL. +In Fig. 7, we show the training of Franka-Reach with +PPO [42] with different frameworks. Although we ensure +same parameters settings for PPO in the frameworks, we +notice a difference in their learning performance and training +time. Since RSL-rl and rl-games are optimized for GPU, we +observe a training speed of 50,000-75,000 frames per second +(FPS) with 2048 environments on an NVIDIA RTX3090. +With stable-baselines3, we receive 6,000-18,0000 FPS. +We also demonstrate training results for different action +spaces in the Franka-Cabinet-Opening task, and var- +ious network architectures and domain randomizations (DR) +in the ShadowHand-Reposing task. In our testing, we +observed that simulation throughput for these environments +are at par with the ones in IsaacGymEnvs [17]. +B. Teleoperation and Imitation Learning +Many manipulation tasks are computationally expensive +or beyond the reach of current RL algorithms. In these + +0.5 +1.0 +1.5 +Steps +×107 +7 +8 +9 +10 +Average Return +PPO on Franka-Reach +Stable Baselines3 +RL Games +RSL RL +0.5 +1.0 +1.5 +2.0 +Steps +×107 +20 +40 +60 +80 +100 +120 +Average Return +RSLRL PPO on Franka-Cabinet-Opening +Joint, position +Joint, velocity +0.5 +1.0 +1.5 +2.0 +Steps +×107 +0 +10 +20 +30 +40 +Consecutive Successes +RLGames PPO on ShadowHand-Repose +Full-State Feed Forward (FF) +Asymmetric actor-critic (AC) FF +Asymmetric AC-FF with DR +Asymmetric AC-LSTM +Asymmetric AC-LSTM with DR +Fig. 7: Franka-Reach is trained with joint position action space using PPO from Stable Baseline3, RL Games, and RSL RL. +Franka-Cabinet-Opening is trained with PPO using different controllers. ShadowHand-Repose for in-hand manipulation of +a cube is trained using variants of PPO with different randomizations, observations, and network types. We evaluate over five seeds and +plot the mean and one standard deviation of the average reward. +Fig. 8: Interactive grasp and motion planning demonstration using ORBIT. The World comprises of objects for table-top manipulation. +The user can select an object from the GUI to grasp. This triggers an image-based grasp generator and allows previewing of the generated +grasps and the robot motion sequence. The user can then choose the grasp and execute the motion on the robot. +TABLE II: Evaluation of policies obtained from behavior cloning +on Franka-Block-Lift environment in the same setting (No +Change), changing initial states (I), goal states (G), and changing +both initial and goal states (Both). We report the the success rate +and trajectory lengths obtained over 100 trials. +Algorithm +Average Traj. Len +Succ. Rate +Eval. Setup +BC +234 +1.00 +No Change +307 +0.89 +G +321 +0.47 +I +324 +0.43 +Both +BC-RNN +249 +1.00 +No Change +251 +1.00 +G +286 +0.88 +I +293 +0.87 +Both +scenarios, boostrapping from user demonstrations provides a +viable path to skill learning. ORBIT provides a data collection +interface that is useful for interacting with the provided +environments using I/O devices and collect data similar to +roboturk [43]. We also provide an interface robomimic [21] +for training imitation learning models. +As +an +example, +we +show +LfD +for +the +Franka-Block-Lift +task. +For +each +of +the +four +settings of initial and desired object positions (fixed or +random start and desired positions), we collect 2000 +trajectories. Using these demonstrations, we train policies +using Behavior Cloning (BC) and BC with an RNN policy +(BC-RNN). We show the performance at test time on 100 +trials in Table II. +C. Motion planning +Motion planning is one of the well-studied domains +in robotics. The traditional Sense-Model-Plan-Act (SMPA) +methodology decomposes the complex problem of reasoning +and control into possible sub-components. ORBIT supports +doing this both procedurally and interactively via the GUI. +a) Hand-crafted policies: We create a state machine +for a given task to perform sequential planning as a separate +node in the agent. It provides the goal states for reaching a +target object, closing the gripper, interacting with the object, +and maneuvering to the next target position. We demonstrate +this paradigm for several tasks in Fig. 6. These hand-crafted +policies can also be utilized for collecting expert demonstra- +tions for challenging tasks such as cloth manipulation. +b) Interactive motion planning: We define a system of +nodes for grasp generation, teleoperation, task-space control, +and motion previewing (shown in Fig. 8). Through the GUI, +the user can select an object to grasp and view the possible +grasp poses and the robot motion sequences generated using +the RMP controller . After confirming the grasp pose, the +robot executes the motion and lifts the object. Following this, +the user obtains teleoperation control of the robot. +D. Deployment on real robot +Deploying an agent on a real robot faces various chal- +lenges, such as dealing with real-time control and safety con- +straints. Different data transport layers, such as ROSTCP [19] +or ZeroMQ (ZMQ) [44], exist for connecting a robotic stack + +FRANKA +THORAFSFRANKA +THOR ATSTHOR ATSTHOR ATSa.1 +a.2 +b.1 +b.2 +Fig. 9: Using simulator as a digital twin to compute and apply commands on the simulated and real robot via ZMQ connection. a) Franka +Panda arm with Allegro hand lifting two objects at once (video). b) Franka Panda performing object avoidance using RMP (video). +1 +2 +3 +Fig. 10: Deployment of an RL policy on ANYmal-D robot using +ROS connection (video). The policy is trained in simulation and +runs at 50 Hz while the actuator net functions at 200 Hz. +to a real platform. We showcase how these mechanisms can +be used with ORBIT to run policies on a real robot. +a) Using +ZMQ: +To +maintain +a +light-weight +and +effiecient communication between, we use ZMQ to send joint +commands from ORBIT to a computer running the real-time +kernel for Franka Emika robot. To abide by the real-time +safety constraints, we use a quintic interpolator to upsample +the 60 Hz joint commands from the simulator to 1000 Hz +for execution on the robot (shown in Fig. 9). +We run experiments on two configurations of the Franka +robot: one with the Franka Emika hand and the other with +an Allegro hand. For each configuration, we showcase three +tasks: 1) teleoperation using a Spacemouse device, 2) de- +ployment of a state machine, and 3) waypoint tracking with +obstacle avoidance. The modular nature of the agent makes +it easy to switch between different control architectures for +each task while using the same interface for the real robot. +b) Using ROS: A variety of existing robots come with +their ROS software stack. In this demonstration, we focus on +how policies trained using ORBIT can be exported and de- +ployed on a robotic platform, particularly for the quadrupedal +robot from ANYbotics, ANYmal-D. +We train a locomotion policy entirely in simulation using +an actuator network [36] for the legged base. To make the +policy robust, we randomize the base mass (22 ± 5 kg) and +add simulated random pushes. We use the contact reporter to +obtain the contact forces and use them in reward design. The +learned policy is deployed on the robot using the ANYmal +ROS stack, (Fig. 10). This sim-to-real transfer indicates the +viability of the simulated contact dynamics and its suitability +for contact-rich tasks in ORBIT. +VI. DISCUSSION +In this paper, we proposed ORBIT: a framework to sim- +plify environment design, enable easier task specifications +and lower the barrier to entry into robotics and robot learn- +ing. ORBIT builds on state-of-the-art physics and render- +ing engines, and provides interfaces to easily design novel +realistic environments comprising various robotic platforms +interacting with rigid and deformable objects, physics-based +sensor simulation and sensor noise models, and different +actuator models. We readily support a broad set of robotic +platforms, ranging from fixed-arm to legged mobile manip- +ulators, CPU and GPU-based motion generators, and object +datasets (such as YCB and Partnet-Mobility). +The breadth of environments possible, as demonstrated +in part in Sec. IV, makes ORBIT useful for broad set of +research questions in robotics. Keeping modularity at its +core, we demonstrated the framework’s extensibility to dif- +ferent paradigms, including reinforcement learning, imitation +learning, and motion planning. We also showcased the ability +to interface the framework to the Franka Emika Panda robot +via ZMQ-based message-passing and sim-to-real deployment +of RL policies for quadrupedal locomotion. +By open-sourcing this framework1, we aim to reduce the +overhead for developing new applications and provide a +unified platform for future robot learning research. While +we continue improving and adding more features to the +framework, we hope that researchers contribute to making +it a one-stop solution for robotics research. +VII. FUTURE WORK +ORBIT can notably simulate physics at up to 125,000 +samples per second; however, camera rendering is currently +bottlenecked to a total of 270 frames per second for ten +cameras rendering 640×480 RGB images on an RTX 3090. +While this number is comparable to other frameworks, we +are actively improving it further by leveraging GPU-based +acceleration for training for visuomotor policies. +1NVIDIA Isaac Sim is free with an individual license. ORBIT will be +open-sourced, and available at https://isaac-orbit.github.io. + +CHEEZLIT +THORLABSCHEEZIT +THORLABS +.QHEEZIT +THORLAIS +THORAISCHEEZIT +THORLATS +THORLATSCHEEZ-T +ORIGiNal +THORLATS +THORATSTHORLAIS +THORATISCHEEZ-IT +ORiGiNal +THORLABSD +OR +THORLABSAdditionally, though our experiments showcase the fidelity +of rigid-contact modeling, the accuracy of contacts in de- +formable objects simulation is still unexplored. It is essential +to note that until now, robot manipulation research in this +domain has not relied on sim-to-real since existing solvers +are typically fragile or slow. Using FEM-based solvers and +physically-based rendering, we believe our framework will +help answer these open questions in the future. +ACKNOWLEDGMENT +We thank Farbod Farshidian for helping with OCS2, Umid +Targuliyev for assisting with imitation learning experiments, +as well as Ossama Samir Ahmed, Lukasz Wawrzyniak, Avi +Rudich, Bryan Peele, Nathan Ratliff, Milad Rakhsha, Vik- +tor Makoviychuk, Jean-Francois Lafleche, Yashraj Narang, +Miles Macklin, Liila Torabi, Philipp Reist, Adam Mora- +vansky, and other members of the NVIDIA PhysX and +Omniverse teams for their assistance with the simulator. +REFERENCES +[1] A. Kumar, Z. Fu, et al., “Rma: Rapid motor adaptation for legged +robots,” Robotics: Science and Systems (RSS), 2021. +[2] Z. Xie, X. Da, et al., “Glide: Generalizable quadrupedal locomotion +in diverse environments with a centroidal model,” arXiv preprint +arXiv:2104.09771, 2021. +[3] T. Miki, J. 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Khatib, “Inertial properties in robotic manipulation: An object- +level framework,” The International Journal of Robotics Research, +vol. 14, no. 1, 1995. +[39] N. Rudin, D. Hoeller, et al., “Learning to walk in minutes using +massively parallel deep reinforcement learning,” in Conference on +Robot Learning (CoRL). +PMLR, 2022. +[40] D. Makoviichuk and V. Makoviychuk, “rl-games: A high-performance +framework for reinforcement learning,” https://github.com/Denys88/rl +games, May 2022. +[41] A. Raffin, A. Hill, et al., “Stable-baselines3: Reliable reinforcement +learning implementations,” Journal of Machine Learning Research, +vol. 22, no. 268, 2021. +[42] J. Schulman, F. Wolski, et al., “Proximal policy optimization algo- +rithms,” arXiv preprint arXiv:1707.06347, 2017. +[43] A. Mandlekar, Y. Zhu, et al., “Roboturk: A crowdsourcing platform +for robotic skill learning through imitation,” in Conference on Robot +Learning (CoRL). +PMLR, 2018. +[44] P. Hintjens, ZeroMQ: messaging for many applications. +” O’Reilly +Media, Inc.”, 2013. + diff --git a/3dE2T4oBgHgl3EQf6Ago/content/tmp_files/load_file.txt b/3dE2T4oBgHgl3EQf6Ago/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9853b9a1567ed1fc63032579af19009bfbb1490 --- /dev/null +++ b/3dE2T4oBgHgl3EQf6Ago/content/tmp_files/load_file.txt @@ -0,0 +1,584 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf,len=583 +page_content='ORBIT: A Unified Simulation Framework for Interactive Robot Learning Environments Mayank Mittal1,2, Calvin Yu3, Qinxi Yu3, Jingzhou Liu1,3, Nikita Rudin1,2, David Hoeller1,2, Jia Lin Yuan3, Pooria Poorsarvi Tehrani3, Ritvik Singh1,3, Yunrong Guo1, Hammad Mazhar1, Ajay Mandlekar1, Buck Babich1, Gavriel State1, Marco Hutter2, Animesh Garg1,3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 1: ORBIT framework provides a large set of robots, sensors, rigid and deformable objects, motion generators, and teleoperation interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Through these, we aim to simplify the process of defining new and complex environments, thereby providing a common platform for algorithmic research in robotics and robot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Abstract— We present ORBIT, a unified and modular frame- work for robot learning powered by NVIDIA Isaac Sim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' It offers a modular design to easily and efficiently create robotic environments with photo-realistic scenes and fast and accurate rigid and deformable body simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' With ORBIT, we provide a suite of benchmark tasks of varying difficulty– from single- stage cabinet opening and cloth folding to multi-stage tasks such as room reorganization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' To support working with diverse observations and action spaces, we include fixed-arm and mobile manipulators with different physically-based sensors and motion generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' ORBIT allows training reinforcement learning policies and collecting large demonstration datasets from hand-crafted or expert solutions in a matter of minutes by leveraging GPU-based parallelization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' In summary, we offer an open-sourced framework that readily comes with 16 robotic platforms, 4 sensor modalities, 10 motion generators, more than 20 benchmark tasks, and wrappers to 4 learning libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' With this framework, we aim to support various research areas, including representation learning, reinforcement learning, imi- tation learning, and task and motion planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We hope it helps establish interdisciplinary collaborations in these communities, and its modularity makes it easily extensible for more tasks and applications in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' For videos, documentation, and code: https://isaac-orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='io/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' INTRODUCTION The recent surge in machine learning has led to a paradigm shift in robotics research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Methods such as reinforcement learning (RL) have shown incredible success in challenging problems such as quadrupedal locomotion [1], [2], [3] and in-hand manipulation [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' However, learning techniques 1 NVIDIA, 2 ETH Z¨urich, 3 University of Toronto, Vector Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Correspondence: mittalma@ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='ch, garg@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='toronto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' require a wealth of training data, which is often challenging and expensive to obtain at scale on a physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' This makes simulators an appealing alternative for developing systems safely, efficiently, and cost-effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' An ideal robot simulation framework needs to provide fast and accurate physics, high-fidelity sensor simulation, diverse asset handling, and easy-to-use interfaces for integrating new tasks and environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' However, existing frameworks often make a trade-off between these aspects depending on their target application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' For instance, simulators designed mainly for vision, such as Habitat [18] or ManipulaTHOR [16], offer decent rendering but simplify low-level interaction intricacies such as grasping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' On the other hand, physics simulators for robotics, such as IsaacGym [17] or Sapien [15], provide fast and reasonably accurate rigid-body contact dynamics but do not include physically-based rendering (PBR), deformable objects simulation or ROS [19] support out-of-the-box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' In this work, we present ORBIT an open-source frame- work, built on NVIDIA Isaac Sim [20], for intuitive designing of environments and tasks for robot learning with photo- realistic scenes and state-of-the-art physics simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Its modular design supports various robotic applications, such as reinforcement learning (RL), learning from demonstrations (LfD), and motion planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Through careful design of inter- faces, we aim to support learning for a diverse range of robots and tasks, allowing operation at different levels of observa- tion (proprioception, images, pointclouds) and action spaces (joint space, task space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' To ensure high-simulation through- put, we leverage hardware-accelerated robot simulation, and include GPU implementations for motion generation and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='04195v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='RO] 10 Jan 2023 TABLE I: Comparison between different simulation frameworks and ORBIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The check (✓) and cross (X) denote presence or absence of the feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' In Robotic Platforms column, M stands for manipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' In Scene Authoring column, G stands for game-based designing, M for mesh-scan scenes, and P for procedural-generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Vectorization Supported Dynamics Sensors Robotic Platforms Name Physics Engine Renderer CPU GPU Rigid Cloth Soft Fluid PBR Tracing RGBD Semantic LiDAR Contact Fixed-M Mobile-M Legged Scene Authoring MetaWorld [6] MuJoCo OpenGL ✓ X ✓ X X X X X X X X ✓ X X P RoboSuite [7] MuJoCo OpenGL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' OptiX ✓ X ✓ X X X X ✓ ✓ X ✓ ✓ X X P DoorGym [8] MuJoCo Unity X X ✓ X X X ✓ ✓ ✓ X X ✓ X X P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' G DEDO [9] Bullet OpenGL ✓ X ✓ ✓ ✓ X X ✓ X X X ✓ X X P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' G RLBench [10] Bullet/ODE OpenGL X X ✓ X X X X ✓ ✓ X ✓ ✓ X X P iGibson [11] Bullet MeshRenderer ✓ X ✓ X X X ✓ ✓ ✓ ✓ X X ✓ X M Habitat 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='0 [12] Bullet Magnum X X ✓ X X X ✓ ✓ ✓ X X ✓ ✓ ✓ P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' M SoftGym [13] FleX OpenGL ✓ X ✓ ✓ ✓ ✓ X X X X X ✓ X X P ThreeDWorld [14] PhysX 4/FleX/Obi Unity3D X X ✓∗ ✓∗ ✓∗ ✓∗ ✓ ✓ ✓ X X X ✓ X P SAPIEN [15] PhysX 4 OptiX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Kuafu ✓ X ✓ X X X ✓ ✓ ✓ X ✓ ✓ ✓ X P ManipulatorThor [16] PhysX 4 Unity X X ✓ X X X ✓ ✓ ✓ X X X ✓ X P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' G IsaacGymEnvs [17] PhysX 5 Vulkan ✓ ✓ ✓ X X X X ✓ ✓ X ✓ ✓ X ✓ P ORBIT (ours) PhysX 5 Omniverse RTX ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' G ThreeDWorld supports simulation of rigid bodies and deformable bodies based on whether PhysX 4 or FleX/Obi is enabled respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Thus, it is limited in simulating interactions between rigid and deformable bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' observations processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' This allows training and evaluation of a complete robotic system at scale, without abstracting out low-level details in robot-environment interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The release of ORBIT v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='0 features: 1) models for three quadrupeds, seven robotic arms, four grippers, two hands, and four mobile manipulators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 2) a selection of CPU and GPU-based motion generators implementations for each robot category, including pre- trained locomotion policies, inverse kinematics, opera- tional space control, and model predictive control;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 3) utilities for collecting human demonstrations using pe- ripherals (keyboard, gamepad or 3D mouse), replaying demonstration datasets, and utilizing them for learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 4) a suite of standardized tasks of varying complexity for benchmark purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' These include eleven rigid object manipulation, thirteen deformable object manipulation, and two locomotion environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Within each task, we allow switching robots, objects, and sensors easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' In the remaining of the paper, we describe the underlying simulation choices (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' II), the framework’s design deci- sions and abstractions (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' III), and its highlighted features (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We demonstrate the framework’s applicability for different workflows (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' V) – particularly RL using various libraries, LfD with robomimic [21], motion planning [22], [23], and connection to physical robots for deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' RELATED WORK Recent years have seen several simulation frameworks, each specializing for particular robotic applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' In this section, we highlight the design choices crucial for building a unified simulation platform and how ORBIT compares to other frameworks (also summarized in Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' a) Physics Engine: Increasing the complexity and re- alism of physically simulated environments is essential for advancing robotics research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' This includes improving the contact dynamics, having better collision handling for non- convex geometries (such as threads), stable solvers for de- formable bodies, and high simulation throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Prior frameworks [7], [10] using MuJoCo [24] or Bul- let [25] focus mainly on rigid object manipulation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Since their underlying physics engines are CPU-based, they need CPU clusters to achieve massive parallelization [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' On the other hand, frameworks for deformable bodies [9], [13] mainly employ Bullet [25] or FleX [26], which use particle- based dynamics for soft bodies and cloth simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' How- ever, limited tooling exists in these frameworks compared to those for rigid object tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' ORBIT aims to bridge this gap by providing a robotics framework that supports rigid and deformable body simulation via PhysX SDK 5 [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' In contrast to other engines, it features GPU-based hardware acceleration for high throughput, signed-distance field (SDF) collision checking [28], and more stable solvers based on finite elements for deformable body simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' b) Sensor simulation: Various existing frameworks [7], [10], [17] use classic rasterization that limits the photo- realism in the generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Recent techniques [29], [30] simulate the interaction of rays with object’s textures in a physically correct manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' These methods helps capture fine visual properties such as transparency and reflection, thereby are promising for bridging sim-to-real visual domain gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' While recent frameworks [15], [12], [16] include physically- based renderers, they mainly support camera-based sen- sors (RGB, depth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' This is insufficient for certain mobile robot applications that need range sensors, such as LiDARs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Leveraging the ray-tracing technology in NVIDIA Isaac Sim, ORBIT supports all these modalities and includes APIs to obtain additional information such as semantic annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' c) Scene designing and asset handling: Frameworks support scene creation procedurally [6], [7], [15], via mesh scans [11], [12] or through game-engine style interfaces [31], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' While mesh scans simplify generating large amounts of scenes, they often suffer from geometric artifacts and lighting problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' On the other hand, procedural generation allows leveraging object datasets for diverse scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' To not restrict to either possibility, we facilitate scene designing by using graphical interfaces and also providing tools for importing different datasets [32], [33], [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Simulators are typically general-purpose and expose ac- cess to various internal properties, often alienating non- expert users due to a steep learning curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' ORBIT inherits many utilities from the NVIDIA Omniverse and Isaac Sim platforms, such as high-quality rendering, multi-format asset import, ROS support, and domain randomization (DR) tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' However, its contributions lie in the specialization of inter- faces for robot learning that simplify environment designing and facilitate transfer to a real robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' For instance, we provide unified abstractions for different robot and object types, allow Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 3: ORBIT’s abstractions comprise World, analogous to the real world, and Agent, the computation graph behind the embodied system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The nodes in the agent’s graph can perform observation-based or action-based processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Through a graph-cut over this computation graph and specifying an extrinsic goal, it is feasible to design different tasks within the same World instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' injecting actuator models into the simulation to assist in sim-to-real transfer, and support various peripherals for data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Overall, it provides a highly featured state-of-the- art simulation framework (Table I) while preserving usability through intuitive abstractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' ORBIT: ABSTRACTIONS AND INTERFACES DESIGN At a high level, the framework design comprises a world and an agent, similar to the real world and the software stack running on the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The agent receives raw observa- tions from the world and computes the actions to apply on the embodiment (robot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Typically in learning, it is assumed that all the perception and motion generation occurs at the same frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' However, in the real world, that is rarely the case: (1) different sensors tick at differing frequencies, (2) depending on the control architecture, actions are applied at different time-scales [35], and (3) various unmodeled sources cause delays and noises in the real system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' In ORBIT, we carefully design the interfaces and abstractions to support (1) and (2), and for (3), we include implementation of different actuator and noise models as part of the robot and sensors respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' a) World: Analogous to the real world, we define a world where robots, sensors, and objects (static or dynamic) exist on the same stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The world can be de- signed procedurally (script-based), via scanned meshes [33], [32], through the game-based GUI of Isaac Sim, or a combination of them, such as importing scanned meshes and adding objects to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' This flexibility reaps the benefits of 3D reconstructed meshes, which capture various archi- tectural layouts, with game-based designing, that simplifies the experience of creating and verifying the scene physics properties by playing the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Robots are a crucial component of the world since they serve as the embodiment for interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' They consist of an articulated system, sensors, and low-level controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The robot class loads its model from USD files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' It may DC Motor Actuator Net (MLP/LSTM) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 4: Illustration of actuator groups for a legged mobile manipu- lator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' This allows decomposing a complex system into sub-groups and defining specific transmission models for each of them flexibly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' have onboard sensors specified through the same USD file or configuration files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The low-level controller processes input actions through the configured actuator models and applies desired joint position, velocity, or torque commands to the simulator (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The actuator dynamics can be modeled using first-principle from physics or be learned as neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' This allows injection of real world actuator characteristics into simulation thereby facilitating sim-to-real transfer of control policies [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Sensors may exist both on the articulation (as part of the robot) or externally (such as, third-person cameras).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' ORBIT interface unifies different physics-based (range, force, and contact sensor) and rendering-based (RGB, depth, normals) sensors under a common interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' To simulate asynchronous sensing and actuation, each sensor has an internal timer that governs its operating frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The sensor only reads the simulator buffers at the configured frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Between the timesteps, the sensor returns the previously obtained values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Objects are passive entities in the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' While several objects may exist in the scene, the user can define objects of interest for a specified task and retrieve data/properties only for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Object properties mainly comprise visual and collision meshes, textures, and physics materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' For any given object, we support randomization of its textures and physics properties, such as friction and joint parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' rt Learning Learning Task Logic Task Agent Rewards/Costs Oracle Reset World Agent Sensors Ot Node 1 Passive Camera External Sensors Objects Node 2 LiDAR Robot at .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Node n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Actuator Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Height Scan ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Visualization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='On-board ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Sensors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Markers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Contact Report ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Computation Nodes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='O NVIDIA Isaac Sim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Motion Generation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Perception ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Filtering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Learning-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='PhysX ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='NVIDIA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='可 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='USD ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Mapping ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Model-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Iray ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='by NVIDIAGripper ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='open/close ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='joint velocity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Mimic Group ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='(6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Actions ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='joint position ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Arm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='joint torque ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='DC Motor ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='(6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='(6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Base ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='joint position ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Actuator Net ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='joint torque ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='(12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='(MLP/LSTM) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='(12)Rigid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Articulated ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Deformable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Cloth ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='IK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='OSC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='RMPFlow ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='OCS2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='NN Policy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Teleoperation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='End-Effector ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Arm ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Mobile Base ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Height Scan ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Camera ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Contact Reporter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Proprioception ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 5: Overview of features included in ORBIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We provide models of different sensors, robotic platforms, objects from different datasets, motion generators and teleoperation devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Using RTX-accelerated ray-tracing, we can obtain high-fidelity images in real-time for different modalities such as RGB, depth, surface normal, instance and semantic segmentation (pixel-wise and bounding boxes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' b) Agent: An agent refers to the decision-making process (“intelligence”) guiding the embodied system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' While roboticists have embraced the modularity of ROS [19], most robot learning frameworks often focus only on the environ- ment definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' This practice requires code replication, & adds friction to switching between different implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Keeping modularity at its core, an agent in ORBIT com- prises various nodes that formulate a computation graph exchanging information between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Broadly, we consider nodes are of two types: 1) perception-based i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=', they process inputs into another representation (such as RGB-D image to point-cloud/TSDF), or 2) action-based i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=', they process in- puts into action commands (such as task-level commands to joint commands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Currently, the flow of information between nodes happens synchronously via Python, which avoids the data exchange overhead of service-client protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' c) Learning task and agent: Paradigms such as RL require specification of a task, a world and may include some computation nodes of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The task logic helps specify the goal for the agent, compute metrics (re- wards/costs) to evaluate the agent’s performance, and manage the episodic resets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' With this component as a separate mod- ule, it becomes feasible to use the same world definition for different tasks, similar to learning in the real world, where tasks are specified through extrinsic reward signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The task definition may also contain different nodes of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' An intuitive way to formalize this is by considering that learning for a particular node happens through a graph cut on the agent’s computation graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' To further concretize the design motivation, consider the example of learning over task space instead of low-level joint actions for lifting a cube [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' In this case, the task-space controller, such as inverse kinematics (IK), would typically run at 50Hz, while the joint controller requires commands at 1000 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Although the task-space controller is a part of the agent’s and not the world’s computation, it is possible to encapsulate that into the task design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' This functionality easily allows switching between motion generators, such as IK, operational-space control (OSC), or reactive planners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' ORBIT: FEATURES While various robotic benchmarks have been proposed [9], [6], [10], the right choice of necessary and sufficient tasks to demonstrate “intelligent” behaviors remains an open ques- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Instead of being prescriptive about tasks, we provide ORBIT as a platform to easily design new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' To facilitate the same, we include a diverse set of supported robots, peripheral devices, and motion generators and a large set of tasks for rigid and soft object manipulation for essential skills such as folding cloth, opening the dishwasher, and screwing a nut into a bolt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Each task showcases aspects of physics and renderer that we believe will facilitate answering crucial research questions, such as building representations for deformable object manipulation and learning skills that generalize to different objects and robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' a) Robots: We support 4 mobile platforms (one om- nidirectional drive base and three quadrupeds), 7 robotic arms (two 6-DoF and five 7-DoF), and 6 end-effectors (four parallel-jaw grippers and two robotic hands).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We provide tools to compose different combinations of these articulations into a complex robotic system such as a legged mobile manipulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' This provides a large set of robot platforms, each of which can be switched in the World.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' b) I/O Devices: Devices define the interface to periph- eral controllers that teleoperate the robot in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The interface reads the input commands from an I/O device and parses them into control commands for subsequent nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' This helps not only in collecting demonstrations [21] but also in debugging the task designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Currently, we include support for Keyboard, Gamepad (Xbox controller), and Spacemouse from 3Dconnexion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' c) Motion Generators: Motion generators transform high-level actions into lower-level commands by treating input actions as reference tracking signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' For instance, inverse kinematics (IK) [37] interprets commands as the desired end-effector poses and computes the desired joint positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Employing these controllers, particularly in task space, has shown to help sim-to-real transferability of robot manipulation policies [7], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' With ORBIT, we include GPU-based implementations for: differential IK [37], operational-space control [38] and joint- level control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Additionally, we provide CPU implementa- tion of state-of-the-art model-based planners such as RMP- Flow [22] for fixed-arm manipulators and OCS2 [23] for whole-body control of mobile manipulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We also provide pre-trained policies for legged locomotion [39] to facilitate solving navigation tasks using base velocity commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=" 40%MORE French's YELLOL3DconnexionF1 F4 F6 F7 F8 SYGR Lock Bresk 2 3 6 7 8 Q R T Home Pgup Cops Lock G H Enter Doier Booe Z X tsift B tshint Pon Ente Alt Alt Ctrt11GPU IK OSC NN PolicyCPU OCS2 RMPF1owated Fluid ClotlTeleoperationFig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 6: Demonstration of the designed tasks using hand-crafted state machines and task-space controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Leveraging recent advances in physics engines, we support high-fidelity simulation of rigid and deformable objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We include environments that allow switching between robots, objects, observations, and action spaces through configuration files (Task videos).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' d) Rigid-body Environments: For rigid-body environ- ments, it is vital to have accurate contact physics, fast collision checking, and articulate joints simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' While some of these tasks exist in prior works [6], [10], [28], [39], we enhance them with our framework’s interfaces and provide more variability using DR tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We also extend ma- nipulation tasks for fixed-arm robots to mobile manipulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' For brevity, we list the environments are as follows: 1) Reach - Track desired pose of the end-effector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 2) Lift - Take an object to a desired position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 3) Beat the Buzz - Displace a key around a pole without touching the pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 4) Nut-Bolt - Tighten a nut on a given bolt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 5) Cabinet - Open or close a cabinet (articulated object).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 6) Pyramid Stack - Stack blocks into pyramids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 7) Hockey [10] - Shoot a puck into the net using a stick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 8) Peg In Hole - Insert blocks into their holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 9) Jenga [10] - Remove and stack blocks into a tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 10) In-Hand Repose - Using dexterous robotic hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 11) Velocity Locomotion - Track a desired velocity command via a legged robot on various terrains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' e) Deformable-body Environments: Deformable ob- jects have a high dimensional state and complex dynamics which are difficult to capture succinctly for robot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' With ORBIT, we provide seventeen deformable objects assets (such as toys and garments) with valid physics configurations and methods to generate new assets (such as rectangular cloth) procedurally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' A concise list of included environments are as follows: 1) Cloth Lifting - Lift a cloth to a target position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 2) Cloth Folding - Fold a cloth into a desired state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 3) Cloth Spreading - Spread a cloth on a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 4) Cloth Dropping - Drop a cloth into a container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 5) Flag Hoisting - Hoist a flag standing on a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 6) Soft Lifting - Lift a soft object to a target position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 7) Soft Placing - Place a soft object on a shelf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 8) Soft Stacking - Stack soft objects on each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 9) Soft Dropping - Drop soft objects into a container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 10) Tower of Hanoi - Stack toruses around a pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 11) Rope Reshaping - Reshape a rope on a table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 12) Fluid Pouring - Pour fluid into another container.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 13) Fluid Transport - Move a filled container without causing any spillages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' It should be noted that the environments (1), (2), and (3) carry the same World definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' They only differ in their task logic module, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' the associated reward associ- ated, which is defined through configuration managers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' This modularity allows code reusage and makes it easier to define a large set of tasks within the same World.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' EXEMPLAR WORKFLOWS WITH ORBIT ORBIT is a unified simulation infrastructure that provides both pre-built environments and easy-to-use interfaces that enables extendability and customization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Owing to high- quality physics, sensor simulation, and rendering, ORBIT us useful for multiple robotics challenges in both perception and decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We outline a subset of such use cases through exemplar workflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' GPU-based Reinforcement Learning We provide wrappers to different RL frameworks (rl- games [40], RSL-rl [39], and stable-baselines-3 [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' This allows users to test their environments on a larger set of RL algorithms and facilitate algorithmic developments in RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 7, we show the training of Franka-Reach with PPO [42] with different frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Although we ensure same parameters settings for PPO in the frameworks, we notice a difference in their learning performance and training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Since RSL-rl and rl-games are optimized for GPU, we observe a training speed of 50,000-75,000 frames per second (FPS) with 2048 environments on an NVIDIA RTX3090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' With stable-baselines3, we receive 6,000-18,0000 FPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We also demonstrate training results for different action spaces in the Franka-Cabinet-Opening task, and var- ious network architectures and domain randomizations (DR) in the ShadowHand-Reposing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' In our testing, we observed that simulation throughput for these environments are at par with the ones in IsaacGymEnvs [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Teleoperation and Imitation Learning Many manipulation tasks are computationally expensive or beyond the reach of current RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' In these 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='5 Steps ×107 7 8 9 10 Average Return PPO on Franka-Reach Stable Baselines3 RL Games RSL RL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='0 Steps ×107 20 40 60 80 100 120 Average Return RSLRL PPO on Franka-Cabinet-Opening Joint, position Joint, velocity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='0 Steps ×107 0 10 20 30 40 Consecutive Successes RLGames PPO on ShadowHand-Repose Full-State Feed Forward (FF) Asymmetric actor-critic (AC) FF Asymmetric AC-FF with DR Asymmetric AC-LSTM Asymmetric AC-LSTM with DR Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 7: Franka-Reach is trained with joint position action space using PPO from Stable Baseline3, RL Games, and RSL RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Franka-Cabinet-Opening is trained with PPO using different controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' ShadowHand-Repose for in-hand manipulation of a cube is trained using variants of PPO with different randomizations, observations, and network types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We evaluate over five seeds and plot the mean and one standard deviation of the average reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 8: Interactive grasp and motion planning demonstration using ORBIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The World comprises of objects for table-top manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The user can select an object from the GUI to grasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' This triggers an image-based grasp generator and allows previewing of the generated grasps and the robot motion sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The user can then choose the grasp and execute the motion on the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' TABLE II: Evaluation of policies obtained from behavior cloning on Franka-Block-Lift environment in the same setting (No Change), changing initial states (I), goal states (G), and changing both initial and goal states (Both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We report the the success rate and trajectory lengths obtained over 100 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Algorithm Average Traj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Len Succ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Rate Eval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Setup BC 234 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='00 No Change 307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='89 G 321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='47 I 324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='43 Both BC-RNN 249 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='00 No Change 251 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='00 G 286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='88 I 293 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='87 Both scenarios, boostrapping from user demonstrations provides a viable path to skill learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' ORBIT provides a data collection interface that is useful for interacting with the provided environments using I/O devices and collect data similar to roboturk [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We also provide an interface robomimic [21] for training imitation learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' As an example, we show LfD for the Franka-Block-Lift task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' For each of the four settings of initial and desired object positions (fixed or random start and desired positions), we collect 2000 trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Using these demonstrations, we train policies using Behavior Cloning (BC) and BC with an RNN policy (BC-RNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We show the performance at test time on 100 trials in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Motion planning Motion planning is one of the well-studied domains in robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The traditional Sense-Model-Plan-Act (SMPA) methodology decomposes the complex problem of reasoning and control into possible sub-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' ORBIT supports doing this both procedurally and interactively via the GUI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' a) Hand-crafted policies: We create a state machine for a given task to perform sequential planning as a separate node in the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' It provides the goal states for reaching a target object, closing the gripper, interacting with the object, and maneuvering to the next target position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We demonstrate this paradigm for several tasks in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' These hand-crafted policies can also be utilized for collecting expert demonstra- tions for challenging tasks such as cloth manipulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' b) Interactive motion planning: We define a system of nodes for grasp generation, teleoperation, task-space control, and motion previewing (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Through the GUI, the user can select an object to grasp and view the possible grasp poses and the robot motion sequences generated using the RMP controller .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' After confirming the grasp pose, the robot executes the motion and lifts the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Following this, the user obtains teleoperation control of the robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Deployment on real robot Deploying an agent on a real robot faces various chal- lenges, such as dealing with real-time control and safety con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Different data transport layers, such as ROSTCP [19] or ZeroMQ (ZMQ) [44], exist for connecting a robotic stack FRANKA THORAFSFRANKA THOR ATSTHOR ATSTHOR ATSa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='1 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='2 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='1 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 9: Using simulator as a digital twin to compute and apply commands on the simulated and real robot via ZMQ connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' a) Franka Panda arm with Allegro hand lifting two objects at once (video).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' b) Franka Panda performing object avoidance using RMP (video).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 1 2 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 10: Deployment of an RL policy on ANYmal-D robot using ROS connection (video).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The policy is trained in simulation and runs at 50 Hz while the actuator net functions at 200 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' to a real platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We showcase how these mechanisms can be used with ORBIT to run policies on a real robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' a) Using ZMQ: To maintain a light-weight and effiecient communication between, we use ZMQ to send joint commands from ORBIT to a computer running the real-time kernel for Franka Emika robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' To abide by the real-time safety constraints, we use a quintic interpolator to upsample the 60 Hz joint commands from the simulator to 1000 Hz for execution on the robot (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We run experiments on two configurations of the Franka robot: one with the Franka Emika hand and the other with an Allegro hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' For each configuration, we showcase three tasks: 1) teleoperation using a Spacemouse device, 2) de- ployment of a state machine, and 3) waypoint tracking with obstacle avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The modular nature of the agent makes it easy to switch between different control architectures for each task while using the same interface for the real robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' b) Using ROS: A variety of existing robots come with their ROS software stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' In this demonstration, we focus on how policies trained using ORBIT can be exported and de- ployed on a robotic platform, particularly for the quadrupedal robot from ANYbotics, ANYmal-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We train a locomotion policy entirely in simulation using an actuator network [36] for the legged base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' To make the policy robust, we randomize the base mass (22 ± 5 kg) and add simulated random pushes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We use the contact reporter to obtain the contact forces and use them in reward design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The learned policy is deployed on the robot using the ANYmal ROS stack, (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' This sim-to-real transfer indicates the viability of the simulated contact dynamics and its suitability for contact-rich tasks in ORBIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' DISCUSSION In this paper, we proposed ORBIT: a framework to sim- plify environment design, enable easier task specifications and lower the barrier to entry into robotics and robot learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' ORBIT builds on state-of-the-art physics and render- ing engines, and provides interfaces to easily design novel realistic environments comprising various robotic platforms interacting with rigid and deformable objects, physics-based sensor simulation and sensor noise models, and different actuator models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We readily support a broad set of robotic platforms, ranging from fixed-arm to legged mobile manip- ulators, CPU and GPU-based motion generators, and object datasets (such as YCB and Partnet-Mobility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' The breadth of environments possible, as demonstrated in part in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' IV, makes ORBIT useful for broad set of research questions in robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Keeping modularity at its core, we demonstrated the framework’s extensibility to dif- ferent paradigms, including reinforcement learning, imitation learning, and motion planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' We also showcased the ability to interface the framework to the Franka Emika Panda robot via ZMQ-based message-passing and sim-to-real deployment of RL policies for quadrupedal locomotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' By open-sourcing this framework1, we aim to reduce the overhead for developing new applications and provide a unified platform for future robot learning research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' While we continue improving and adding more features to the framework, we hope that researchers contribute to making it a one-stop solution for robotics research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' FUTURE WORK ORBIT can notably simulate physics at up to 125,000 samples per second;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' however, camera rendering is currently bottlenecked to a total of 270 frames per second for ten cameras rendering 640×480 RGB images on an RTX 3090.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' While this number is comparable to other frameworks, we are actively improving it further by leveraging GPU-based acceleration for training for visuomotor policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' 1NVIDIA Isaac Sim is free with an individual license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' ORBIT will be open-sourced, and available at https://isaac-orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='io.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' CHEEZLIT THORLABSCHEEZIT THORLABS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content='QHEEZIT THORLAIS THORAISCHEEZIT THORLATS THORLATSCHEEZ-T ORIGiNal THORLATS THORATSTHORLAIS THORATISCHEEZ-IT ORiGiNal THORLABSD OR THORLABSAdditionally, though our experiments showcase the fidelity of rigid-contact modeling, the accuracy of contacts in de- formable objects simulation is still unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' It is essential to note that until now, robot manipulation research in this domain has not relied on sim-to-real since existing solvers are typically fragile or slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Using FEM-based solvers and physically-based rendering, we believe our framework will help answer these open questions in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' ACKNOWLEDGMENT We thank Farbod Farshidian for helping with OCS2, Umid Targuliyev for assisting with imitation learning experiments, as well as Ossama Samir Ahmed, Lukasz Wawrzyniak, Avi Rudich, Bryan Peele, Nathan Ratliff, Milad Rakhsha, Vik- tor Makoviychuk, Jean-Francois Lafleche, Yashraj Narang, Miles Macklin, Liila Torabi, Philipp Reist, Adam Mora- vansky, and other members of the NVIDIA PhysX and Omniverse teams for their assistance with the simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' REFERENCES [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Kumar, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=' Fu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dE2T4oBgHgl3EQf6Ago/content/2301.04195v1.pdf'} +page_content=', “Rma: Rapid motor adaptation for legged robots,” Robotics: Science and Systems (RSS), 2021.' metadata={'source': 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index 0000000000000000000000000000000000000000..a1de9fdc96ecbe0d4c6a92530e8a2ca39ec26e24 --- /dev/null +++ b/6tA0T4oBgHgl3EQfOP83/content/tmp_files/2301.02157v1.pdf.txt @@ -0,0 +1,970 @@ +Astronomy & Astrophysics manuscript no. main +©ESO 2023 +January 6, 2023 +Letter to the Editor +Asteroids’ reflectance from Gaia DR3: +Artificial reddening at near-UV wavelengths +F. Tinaut-Ruano1, 2, E. Tatsumi1, 2, 3, P. Tanga4, J. de León1, 2, M. Delbo4, F. De Angeli5, D. Morate1, 2, J. Licandro1, 2, +and L. Galluccio4 +1 Instituto de Astrofísica de Canarias (IAC), C/ Vía Láctea, s/n, E-38205, La Laguna, Spain +e-mail: fernando.tinaut@iac.es +2 Department of Astrophysics, University of La Laguna, Tenerife, Spain +3 Department of Earth and Planetary Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, 113-0033 Tokyo, Japan +4 Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304 +Nice Cedex 4, France +5 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK +Received 04/10/2022; accepted 02/01/2023 +ABSTRACT +Context. Observational and instrumental difficulties observing small bodies below 0.5 µm make this wavelength range poorly studied +compared with the visible and near-infrared. Furthermore, the suitability of many commonly used solar analogues, essential in the +computation of asteroid reflectances, is usually assessed only in visible wavelengths, while some of these objects show spectra that +are quite different from the spectrum of the Sun at wavelengths below 0.55 µm. Stars HD 28099 (Hyades 64) and HD 186427 (16 +Cyg B) are two well-studied solar analogues that instead present spectra that are also very similar to the spectrum of the Sun in the +wavelength region between 0.36 and 0.55 µm. +Aims. We aim to assess the suitability in the near-ultraviolet (NUV) region of the solar analogues selected by the team responsible for +the asteroid reflectance included in Gaia Data Release 3 (DR3) and to suggest a correction (in the form of multiplicative factors) to +be applied to the Gaia DR3 asteroid reflectance spectra to account for the differences with respect to the solar analogue Hyades 64. +Methods. To compute the multiplicative factors, we calculated the ratio between the solar analogues used by Gaia DR3 and Hyades +64, and then we averaged and binned this ratio in the same way as the asteroid spectra in Gaia DR3. We also compared both the +original and corrected Gaia asteroid spectra with observational data from the Eight Color Asteroid Survey (ECAS), one UV spectrum +obtained with the Hubble Space Telescope (HST) and a set of blue-visible spectra obtained with the 3.6m Telescopio Nazionale +Galileo (TNG). By means of this comparison, we quantified the goodness of the obtained correction. +Results. We find that the solar analogues selected for Gaia DR3 to compute the reflectance spectra of the asteroids of this data release +have a systematically redder spectral slope at wavelengths shorter than 0.55 µm than Hyades 64. We find that no correction is needed +in the red photometer (RP, between 0.7 and 1 µm), but a correction should be applied at wavelengths below 0.55 µm, that is in the +blue photometer (BP). After applying the correction, we find a better agreement between Gaia DR3 spectra, ECAS, HST, and our set +of ground-based observations with the TNG. +Conclusions. Correcting the near-UV part of the asteroid reflectance spectra is very important for proper comparisons with laboratory +spectra (minerals, meteorite samples, etc.) or to analyse quantitatively the UV absorption (which is particularly important to study +hydration in primitive asteroids). The spectral behaviour at wavelengths below 0.5 µm of the selected solar analogues should be fully +studied and taken into account for Gaia DR4. +Key words. Gaia – asteroids – Solar analogues – UV – spectra +1. Introduction +Asteroid reflectance spectra and/or spectrophotometry pro- +vide(s) information on their surfaces’ composition and the pro- +cesses that modify their properties such as space weathering +(Reddy et al. 2015). Historically, the use of photoelectric detec- +tors (or photometers), which are more sensitive at bluer wave- +lengths (e.g. < 0.5 µm), and the development of the standard +UBV photometric system (Johnson & Morgan 1951) led to the +appearance of the first asteroid taxonomies in the 1970s (Zell- +ner 1973; Chapman et al. 1975), which contained information +at blue-visible wavelengths or what we call near-UV (NUV). +The introduction of the charge-coupled-devices (CCDs) in as- +tronomy in the 1990s and later on contributed to the ’loss’ of +NUV information, as CCDs were much less sensitive at those +wavelengths. Therefore, the large majority of the modern spec- +troscopic and spectrophotometric surveys cover the wavelength +range from ∼0.5 µm up to 2.5 µm. Nevertheless, there are some +exceptions. One of the first large surveys with information in the +NUV is the Eight Asteroid Survey (ECAS, Zellner et al. 1985). +In this survey, we can find the photometry in eight broad-band +filters between 0.34 to 1.04 µm for 589 minor planets, includ- +ing two filters below 0.45 µm. These observations were used to +develop a new taxonomy (see Tholen 1984). Other recent cat- +alogues, such as the Sloan Digital Sky Survey (SDSS) Moving +Objects catalogue (Ivezi´c et al. 2002), the Moving Objects Ob- +served from Javalambre (MOOJa) catalogue from the J-PLUS +survey (Morate et al. 2021), or the Solar System objects obser- +vations from the SkyMapper Southern Survey (Sergeyev et al. +Article number, page 1 of 13 +arXiv:2301.02157v1 [astro-ph.EP] 5 Jan 2023 + +A&A proofs: manuscript no. main +2022), also include photometry in five, 12, and six filters be- +tween 0.3 and 1.1 µm with 104,449, 3122, and 205,515 objects +observed, respectively. The new Gaia data release 3 (DR3 here- +after) catalogue, which was released in June 2022, offers 60,518 +objects binned in 16 wavelengths between 0.352 and 1.056 µm +to mean reflectance spectra. +Even though some laboratory measurements suggest the po- +tential of the NUV absorption as a diagnostic region of hydrated +and ferric material (Gaffey & McCord 1979; Feierberg 1981; +Feierberg et al. 1985; Hiroi et al. 1996; Cloutis et al. 2011a,b; +Hendrix et al. 2016; Hiroi et al. 2021), a quantitative distribu- +tion of the NUV absorption among asteroids has not been dis- +cussed before (Tatsumi et al. 2022). The small sensitivity of +CCDs and the lower Sun’s emission in NUV wavelengths make +observations difficult. Moreover, the Rayleigh scattering by the +atmosphere is stronger on shorter wavelengths, decreasing the +signal-to-noise ratio (S/N) for the NUV region observed from +the ground. To compute the reflectance spectra, we needed to di- +vide wavelength by wavelength of the measured spectra by the +spectra of the Sun. As it is unpractical to observe the Sun with +the same instrument used to observe asteroids, we used solar +analogues (SAs), that is stars selected by their known similar +spectra to that of the Sun. As the large majority of the spec- +troscopic and spectrophotometric surveys cover the wavelength +range that goes from the visible to near-infrared (NIR), the most +commonly used SAs are well characterised at those wavelengths +but they can behave very differently in the NUV. This flux dif- +ference at bluer wavelengths can introduce systematic errors in +the asteroid reflectance spectra. A good example is the work by +de León et al. (2016), where they searched for the presence of +F-type asteroids in the Polana collisional family since the parent +body of the family, asteroid (142) Polana, was classified as an +F type. The authors obtained reflectance spectra in the NUV of +the members of the family, finding that the large majority were +classified as B types. As most of the observers, they used SAs +that were widely used by the community. Interestingly, after ob- +taining the asteroid reflectances again using only Hyades 64 as +the SA, Tatsumi et al. (2022) found that the large majority of +the observed members of the Polana family were indeed F types +and not B types. This evidences the importance of using ade- +quate SAs when observing in the NUV, and it has been the main +motivation for this work. +In this Letter, we present a comparison between the SAs se- +lected to compute the reflectance spectra in the frame of the data +processing of Gaia DR3 (Gaia Collaboration et al. 2022) and +Hyades 64. We analyse the results from this comparison and +propose a multiplicative correction that can be applied to the +archived asteroids’ reflectance spectra. We finally tested it by +comparing corrected Gaia reflectance spectra with ground-based +observations that have also been corrected against the same SA +(ECAS survey, TNG spectra) and with one observation with the +Hubble Space Telescope (HST). +2. Sample +2.1. Solar analogues in Gaia DR3 +The Gaia DR3 catalogue (Gaia Collaboration et al. 2022) gives +access to internally and externally calibrated mean spectra for a +large subset of sources. Internally calibrated spectra refer to an +internal reference system that is homogeneous across all differ- +ent instrumental configurations, while externally calibrated spec- +tra are given in an absolute wavelength and flux scale (see De +Angeli et al. 2022; Montegriffo et al. 2022, for more details). +Epoch spectra (spectra derived from a single observation rather +than averaging many observations of the same source) are not +included in this release. For this Letter, we relied on internally +calibrated data when computing the correction for the Gaia re- +flectances to ensure consistency and to avoid artefacts that could +appear when dividing two externally calibrated spectra, as they +are polynomial fits. +To select the SAs, the Gaia team did a bibliographic search +and selected a list of stars that are widely used as solar analogues +for asteroid spectroscopy (Bus & Binzel 2002; Lazzaro et al. +2004; Soubiran & Triaud 2004; Fornasier et al. 2007; Popescu +et al. 2014; Perna et al. 2018; Popescu et al. 2019; Lucas et al. +2019). First of all, we note that the star identified as 16 Cygnus +B in Gaia Collaboration et al. (2022) is in fact 16 Cygnus A and +that the parameters in their Table C.1. correspond to those of 16 +Cygnus A. Luckily enough, the spectrum of 16 Cygnus B was +also available in DR3. Among the referenced works, only Soubi- +ran & Triaud (2004) carried out a search for SAs by comparing +their spectra to that of the Sun down to 0.385 µm. The rest sim- +ply used G2V stars or cited previous works that presented SAs, +as in Hardorp (1978). In this later work, Hardorp selected SAs +by comparing their spectra with the spectrum of the Sun using +wavelengths down to 0.36 µm. He highlighted the variations that +can exist at NUV wavelengths even between stars of the same +spectral class. +2.2. Asteroids in Gaia DR3 +Among the Gaia DR3 products for Solar System objects (SSOs), +neither the internally nor the externally calibrated spectra are +available to the community, as is the case for the stars. This is due +to a specific choice of the Data Processing and Analysis Consor- +tium (DPAC) caused by the difficulty of calculating those quanti- +ties owing to the intrinsic variability and proper motion of SSOs. +Instead, for each SSO and each epoch, the nominal, pre-launch +dispersion function was used to convert pseudo-wavelengths to +physical wavelengths. The reflectance spectra were calculated by +dividing each epoch spectrum by the mean of the SAs selected +and then averaging over the set of epochs. After that, a set of +fixed wavelengths every 44 nm in the range between 374 and +1034 nm was defined, with a set of bins centred at those wave- +lengths and with a size of 44 nm. For each bin (a total of 16 are +provided), a σ-clipping filter was applied and a weighted aver- +age using the inverse of the standard deviation as weight was +obtained. Finally, the reflectances were normalised to unity us- +ing the value at 550±25 nm. This final product is the only one +available in DR3. +2.3. Hyades 64 & 16 Cyg B +As mentioned in Sect. 2.1, Hardorp (1978) concluded that +Hyades 64 and 16 Cyg B are two of the four stars that exhibit +’almost indistinguishable’ NUV spectra (quoting the author’s +words) from the spectrum of the Sun. This was confirmed in +subsequent papers from the same author (Hardorp 1980a,b) and +from other researchers (Cayrel de Strobel 1996; Porto de Mello +& da Silva 1997; Farnham et al. 2000; Soubiran & Triaud 2004). +We used these two stars as a ’reference’ to compute the correc- +tion factor to be applied to the Gaia DR3 asteroids spectra, as +they are in the list of SAs selected by Gaia Collaboration et al. +(2022). The methodology is described in the following section. +We note that the obtained correction factor using Hyades 64 as +opposed to 16 Cygnus B differs less than 0.5%. We, therefore, +Article number, page 2 of 13 + +F. Tinaut-Ruano et al.: Asteroids’ reflectance from Gaia DR3: Artificial reddening at near-UV wavelengths +Table 1. Multiplicative correction factors for Gaia asteroid binned spec- +tra. We include the wavelengths below 0.55 µm. +Wavelength (µm) +Correction factor +0.374 +1.07 +0.418 +1.05 +0.462 +1.02 +0.506 +1.01 +0.550 +1.00 +decided to use Hyades 64, as it was the star that was used for +both the ECAS survey and our ground-based observations. +3. Methodology +3.1. Computing the correction factor: Internally calibrated +data +In order to compute a correction applicable to the Gaia DR3 +reflectances, we proceeded as follows: first, using the internally +calibrated data, we computed the ratio between the Gaia sample +of SAs, as well as the mean spectrum of these SAs, and Hyades +64 (Fig. 1). As we can observe in the right panel of Fig. 1, which +corresponds to the red photometer (RP), the deviation from the +unity of the ratio between Gaia’s mean SA and Hyades 64 (black +line) is always below 1%. Therefore, this mean spectrum can +confidently be used to obtain the reflectance spectra of asteroids +above 0.55 µm. +However, the situation in the the blue photometer (BP) is +quite different. We can see in the left panel of Fig. 1 that the de- +viation from the unity of the above defined ratio can reach values +of up to 10%, indicating that the mean spectrum of the SAs used +in Gaia DR3 differs significantly from Hyades 64 at wavelengths +below 0.55 µm. The biggest effect when using this mean spec- +trum to obtain asteroids’ reflectance spectra is the introduction +of a systematic (and not real) positive slope, in particular in the +range between 0.4 and 0.55 µm, mimicking a drop in reflectance +below 0.55 µm. Furthermore, the division by this mean spec- +trum can also introduce a ’fake’ absorption around 0.38 µm. We +have quantified this spectral slope in two separate wavelength +ranges, trying to reproduce the observed behaviour of the ratio: +one slope between 0.4 to 0.55 µm, which we named S Blue, and +another one for wavelengths below 0.4 µm, named µm S UV. The +obtained values for the individual SAs used in Gaia DR3 (blue +stars), as well as for the mean spectrum (blue cross) are shown +in Fig. 2. For the mean spectrum of the SAs used in Gaia DR3, +we found that the introduced slopes are S Blue = -0.38 µm−1 and +S UV = 0.69 µm−1. +From this analysis, we conclude that a correction is needed +in the NUV wavelengths, that is below 0.55 µm. To arrive at +the multiplicative correction factors, we binned the ratio between +the mean spectra of SAs selected by the DPAC and Hyades 64, +using the same wavelengths and bin size as the ones adopted for +the asteroid reflectance spectra in the Gaia DR3 (see Sect. 2). In +this way, the users can easily correct the asteroid spectra at NUV +wavelengths. The obtained values are shown in Table 1. +3.2. Comparison of corrected reflectances with existing data +To correct the artificial slopes introduced by the use of the mean +Gaia SAs, we multiplied the binned asteroid reflectance spec- +tra below 0.55 µm by the corresponding correction factors. We +compared the corrected Gaia spectra with spectra or spectropho- +tometry of the same asteroids obtained using other facilities. As +a first step, we selected only those Gaia asteroid spectra with +a S/N > 160, as we detected a systematic decrease in spectral +slope values at blue wavelengths with decreasing S/N for objects +with a smaller S/N than 150. We then selected spectrophotomet- +ric data from the ECAS survey for asteroids that have more than +one observation, and NUV spectra obtained with the Telesco- +pio Nazionale Galileo (TNG) and previously published by Tat- +sumi et al. (2022). The resulting comparison dataset is shown in +Fig. A.1, where the red lines correspond to the original Gaia re- +flectances, black lines are the corrected ones, dark blue lines cor- +respond to ECAS data, and TNG spectra are shown in light blue. +As can be seen, the corrected reflectances are in better agree- +ment with the ECAS and TNG data than the original ones. We +also included the UV spectrum of asteroid (624) Hector down- +loaded from the ESA archive using the python package astro- +query.esa.hubble1. It was obtained with STIS at HST (Wong +et al. 2019). We converted the flux to reflectance using the spec- +trum of the Sun provided for the STIS instrument2. We note +that even after the correction, some asteroids show discrepancies +with the reference data. This is discussed in the next section. +4. Results and discussion +We have shown that the artificial slope introduced at blue +wavelengths in the Gaia DR3 asteroid data due to the selected +SAs is -0.38 µm−1 in the range between 0.4 to 0.55 µm and 0.69 +µm−1 below 0.4 µm. Following Zellner et al. (1985), the b and v +filters of the ECAS survey have central effective wavelengths of +0.437 and 0.550 µm, respectively. According to Tholen (1984), +the (b-v) colours of the mean F and B taxonomical classes are +-0.049 and -0.015 magnitudes, respectively. Transforming these +colours to relative reflectances results in 1.046 and 1.014, which +gives slopes of -0.407 and -0.124 µm−1 between 0.437 and 0.55 +µm. Therefore, the difference between these computed slopes +for F and B types (-0.283 µm−1) is smaller than the artificial +slope introduced by the use of the mean SA of Gaia, implying +that unless we apply the correction proposed in this Letter, +asteroids can be easily misclassified as B types when actually +being F types (see the described example in the Introduction for +the case of members of the Polana family). +To test and quantify the goodness of our proposed correction, +we computed the spectral slope between 0.437 and 0.55 µm for +the ECAS comparison dataset, and between 0.418 and 0.55 for +Gaia original and corrected spectra. In Fig. 3 we plotted the +difference between those slopes. After applying our correction +factor, we could see that the large majority (148 out of 152) of +the asteroids have more similar slopes to those of ECAS. +Nevertheless, our correction has limitations. First, we were +testing its goodness over space-based observations using +ground-based observations. For wavelengths down to 0.3 µm, +ground-based observations present some difficulties, mainly due +to the atmospheric absorption and the lower sensitivity of the +detectors. Furthermore, Gaia observations at those wavelengths +also have other artifacts that we do not fully understand, such +as the detected strong decrease in the spectral slope below S/N +150. Another point to consider when comparing asteroid spectra +observed in different epochs is the effect of the different viewing +geometries. This difference in the viewing geometry, and thus, in +1 https://astroquery.readthedocs.io/en/latest/esa/ +hubble/hubble.html +2 https://archive.stsci.edu/hlsps/reference-atlases/ +cdbs/current_calspec/sun_reference_stis_002.fits +Article number, page 3 of 13 + +A&A proofs: manuscript no. main +0.35 +0.40 +0.45 +0.50 +0.55 +0.60 +Wavelength [ m] +0.95 +1.00 +1.05 +1.10 +1.15 +Counts relative to Hyades 64 +BP +0.7 +0.8 +0.9 +1.0 +Wavelength [ m] +RP +HD060234 +HD123758 +16 Cyg B(A)1 +HD6400 +HD220022 +HD220764 +HD016640 +HD292561 +HD100044 +HD155415 +SA110-361 +HD182081 +HD144585 +HD146233 +HD138159 +HD139287 +HD020926 +HD154424 +HD202282 +16 Cyg B +mean +Fig. 1. Ratio between the internally calibrated spectra of each of the Gaia SAs and Hyades 64 in the blue photometer (BP, left panel) and the red +photometer (RP, right panel). We also plotted the ratio of the mean Gaia SA and Hyades 64 (black solid line) and the binned version of this ratio +at the wavelengths provided for SSO in Gaia DR3 (black dots). +1 We note that the star identified as 16 Cygnus B in Gaia Collaboration et al. (2022) is in fact 16 Cyg A (see the main text for more details). +1.0 +0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +SUV [ m +1] +0.8 +0.6 +0.4 +0.2 +0.0 +SBlue [ m +1] +Gaia SAs +Gaia mean +Fig. 2. Slopes introduced by each of the SAs in the Gaia sample (blue +stars) and their mean (blue cross), compared to Hyades 64. We note that +S Blue was computed in the 0.4–0.55 µm range, while S UV was computed +using wavelengths below 0.4 µm. +the phase angle, causes a change in the spectral slope known as +phase reddening or phase coloring Alvarez-Candal et al. (2022). +This effect has not been well studied at blue wavelengths. Still, +even in the event that we were able to correct it, Gaia’s spectra +are, on average, over different epochs and the information on +the phase angle values is not provided. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ECAS slope - original Gaia slope (1/ ) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +ECAS slope - corrected Gaia slope (1/ ) +Fig. 3. Difference between the blue slope for ECAS and for Gaia origi- +nal data (x-axis) and corrected data (y-axis) in the comparison sample. +5. Conclusions +We have found that the use of the SAs selected to compute the +reflectance spectra of the asteroids in Gaia DR3 introduces an +artificial reddening in the spectral slope below 0.5 µm, that is +an artificial drop in reflectance. By comparing those SAs with +Hyades 64, one of the best characterised SAs at NUV wave- +lengths, we obtain multiplicative correction factors for each of +the reflectance wavelengths below 0.55 µm (a total of four) that +can be applied to the asteroids’ reflectance spectra in Gaia DR3. +By applying this correction, we found a better agreement be- +tween the Gaia spectra and other data sources such as ECAS. +Article number, page 4 of 13 + +F. Tinaut-Ruano et al.: Asteroids’ reflectance from Gaia DR3: Artificial reddening at near-UV wavelengths +The behaviour of the SAs in the red wavelengths is in agree- +ment with Hyades 64 within 1%. This was somehow expected, +as the majority of the SAs used by the Gaia team were previ- +ously tested and widely used by the community to obtain visible +reflectance spectra of asteroids, typically beyond 0.45–0.5 µm. +Correcting the NUV part of the asteroid reflectance spectra is +fundamental to study the presence of the UV absorption, which +has been associated with hydration in primitive asteroids, or to +discriminate between B and F types, which are two taxonom- +ical classes that have proven to have very distinct polarimetric +properties. The NUV region has not yet been fully exploited for +asteroids and, in this way, Gaia spectra constitute a major step +forward in our understanding of these wavelengths. +Acknowledgements. FTR, JdL, ET, DM, and JL acknowledge support from the +Agencia Estatal de Investigación del Ministerio de Ciencia e Innovación (AEI- +MCINN) under the grant ’Hydrated Minerals and Organic Compounds in Prim- +itive Asteroids’ with reference PID2020-120464GB-100. +FTR also acknowledges the support from the COST Action and the ESA +Archival Visitor Programme. +DM acknowledges support from the ESA P3NEOI programme (AO/1- +9591/18/D/MRP). +This work has made use of data from the European Space Agency (ESA) +mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia +Data Processing and Analysis Consortium (DPAC, https://www.cosmos. +esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been pro- +vided by national institutions, in particular, the institutions participating in the +Gaia Multilateral Agreement. +The work of MD is supported by the CNES and by the project Origins of the +French National Research Agency (ANR-18-CE31-0014). +F. De Angeli is supported by the United Kingdom Space Agency (UKSA) +through the grants ST/X00158X/1 and ST/W002469/1. +References +Alvarez-Candal, A., Jimenez Corral, S., & Colazo, M. 2022, A&A, 667, A81 +Bus, S. J. & Binzel, R. P. 2002, Icarus, 158, 106 +Cayrel de Strobel, G. 1996, A&A Rev., 7, 243 +Chapman, C. R., Morrison, D., & Zellner, B. 1975, Icarus, 25, 104 +Cloutis, E. A., Hiroi, T., Gaffey, M. J., Alexander, C. M. O. D., & Mann, P. +2011a, Icarus, 212, 180 +Cloutis, E. A., Hudon, P., Hiroi, T., Gaffey, M. J., & Mann, P. 2011b, Icarus, 216, +309 +De Angeli, F., Weiler, M., Montegriffo, P., et al. 2022, arXiv e-prints, +arXiv:2206.06143 +de León, J., Pinilla-Alonso, N., Delbo, M., et al. 2016, Icarus, 266, 57 +Farnham, T. L., Schleicher, D. G., & A’Hearn, M. F. 2000, Icarus, 147, 180 +Feierberg, M. A. 1981, PhD thesis, University of Arizona +Feierberg, M. A., Lebofsky, L. A., & Tholen, D. 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Tinaut-Ruano et al.: Asteroids’ reflectance from Gaia DR3: Artificial reddening at near-UV wavelengths +1.0 +0.8 +1 +ECAS +original Gaia +corrected Gaia +2 +3 +1.0 +0.8 +4 +6 +7 +1.0 +0.8 +8 +9 +10 +1.0 +0.8 +Relative reflectance +12 +14 +16 +1.0 +0.8 +18 +21 +23 +1.0 +0.8 +24 +27 +29 +1.0 +0.8 +37 +38 +39 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +1.0 +0.8 +42 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +44 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +45 +Fig. A.1. Comparison between ground-based observations from the Eight Asteroid Survey (ECAS, dark blue line), TNG observations (light blue +line) original Gaia data (red line), and corrected data (black line). We also included a UV spectrum of asteroid (624) downloaded from ESA +archive and obtained with the instrument STIS, on board the Hubble Space Telescope (HST). +Article number, page 7 of 13 + +A&A proofs: manuscript no. main +1.0 +0.8 +46 +47 +TNG +ECAS +original Gaia +corrected Gaia +51 +1.0 +0.8 +62 +64 +65 +1.0 +0.8 +71 +80 +82 +1.0 +0.8 +Relative reflectance +83 +85 +86 +1.0 +0.8 +87 +88 +90 +1.0 +0.8 +93 +94 +95 +1.0 +0.8 +97 +101 +103 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +1.0 +0.8 +105 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +106 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +107 +Article number, page 8 of 13 + +F. Tinaut-Ruano et al.: Asteroids’ reflectance from Gaia DR3: Artificial reddening at near-UV wavelengths +1.0 +0.8 +109 +111 +114 +1.0 +0.8 +117 +124 +132 +1.0 +0.8 +134 +135 +137 +1.0 +0.8 +Relative reflectance +142 +153 +158 +1.0 +0.8 +168 +171 +179 +1.0 +0.8 +187 +190 +198 +1.0 +0.8 +211 +213 +216 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +1.0 +0.8 +221 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +225 +TNG +ECAS +original Gaia +corrected Gaia +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +229 +Article number, page 9 of 13 + +A&A proofs: manuscript no. main +1.0 +0.8 +233 +236 +261 +TNG +ECAS +original Gaia +corrected Gaia +1.0 +0.8 +268 +275 +279 +1.0 +0.8 +287 +306 +308 +1.0 +0.8 +Relative reflectance +322 +323 +326 +1.0 +0.8 +334 +339 +349 +1.0 +0.8 +354 +361 +368 +1.0 +0.8 +369 +374 +379 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +1.0 +0.8 +380 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +383 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +389 +Article number, page 10 of 13 + +F. Tinaut-Ruano et al.: Asteroids’ reflectance from Gaia DR3: Artificial reddening at near-UV wavelengths +1.0 +0.8 +394 +406 +407 +1.0 +0.8 +419 +TNG +ECAS +original Gaia +corrected Gaia +420 +433 +1.0 +0.8 +434 +442 +443 +1.0 +0.8 +Relative reflectance +470 +471 +480 +1.0 +0.8 +483 +509 +512 +1.0 +0.8 +522 +529 +532 +1.0 +0.8 +558 +566 +570 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +1.0 +0.8 +579 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +602 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +616 +Article number, page 11 of 13 + +A&A proofs: manuscript no. main +1.0 +0.8 +624 +HST +TNG +ECAS +original Gaia +corrected Gaia +635 +639 +1.0 +0.8 +654 +664 +686 +1.0 +0.8 +699 +702 +704 +1.0 +0.8 +Relative reflectance +712 +714 +721 +1.0 +0.8 +733 +739 +748 +1.0 +0.8 +773 +778 +785 +1.0 +0.8 +786 +849 +863 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +1.0 +0.8 +897 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +914 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +931 +Article number, page 12 of 13 + +F. Tinaut-Ruano et al.: Asteroids’ reflectance from Gaia DR3: Artificial reddening at near-UV wavelengths +1.0 +0.8 +980 +ECAS +original Gaia +corrected Gaia +1001 +1021 +1.0 +0.8 +1105 +1144 +1172 +1.0 +0.8 +Relative reflectance +1180 +1266 +1268 +1.0 +0.8 +1275 +1509 +1604 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +1.0 +0.8 +1606 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +1650 +0.35 +0.40 +0.45 +0.50 +0.55 +Wavelength [ m] +1754 +Article number, page 13 of 13 + diff --git a/6tA0T4oBgHgl3EQfOP83/content/tmp_files/load_file.txt b/6tA0T4oBgHgl3EQfOP83/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..62c25084fabbb267633dd94e3f6cdfbc4e5e1bcf --- /dev/null +++ b/6tA0T4oBgHgl3EQfOP83/content/tmp_files/load_file.txt @@ -0,0 +1,786 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf,len=785 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' main ©ESO 2023 January 6, 2023 Letter to the Editor Asteroids’ reflectance from Gaia DR3: Artificial reddening at near-UV wavelengths F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Tinaut-Ruano1, 2, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Tatsumi1, 2, 3, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Tanga4, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' de León1, 2, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Delbo4, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' De Angeli5, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Morate1, 2, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Licandro1, 2, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Galluccio4 1 Instituto de Astrofísica de Canarias (IAC), C/ Vía Láctea, s/n, E-38205, La Laguna, Spain e-mail: fernando.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='tinaut@iac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='es 2 Department of Astrophysics, University of La Laguna, Tenerife, Spain 3 Department of Earth and Planetary Science, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, 113-0033 Tokyo, Japan 4 Université Côte d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice Cedex 4, France 5 Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Received 04/10/2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' accepted 02/01/2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Observational and instrumental difficulties observing small bodies below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='5 µm make this wavelength range poorly studied compared with the visible and near-infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Furthermore, the suitability of many commonly used solar analogues, essential in the computation of asteroid reflectances, is usually assessed only in visible wavelengths, while some of these objects show spectra that are quite different from the spectrum of the Sun at wavelengths below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Stars HD 28099 (Hyades 64) and HD 186427 (16 Cyg B) are two well-studied solar analogues that instead present spectra that are also very similar to the spectrum of the Sun in the wavelength region between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='36 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We aim to assess the suitability in the near-ultraviolet (NUV) region of the solar analogues selected by the team responsible for the asteroid reflectance included in Gaia Data Release 3 (DR3) and to suggest a correction (in the form of multiplicative factors) to be applied to the Gaia DR3 asteroid reflectance spectra to account for the differences with respect to the solar analogue Hyades 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' To compute the multiplicative factors, we calculated the ratio between the solar analogues used by Gaia DR3 and Hyades 64, and then we averaged and binned this ratio in the same way as the asteroid spectra in Gaia DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We also compared both the original and corrected Gaia asteroid spectra with observational data from the Eight Color Asteroid Survey (ECAS), one UV spectrum obtained with the Hubble Space Telescope (HST) and a set of blue-visible spectra obtained with the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='6m Telescopio Nazionale Galileo (TNG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' By means of this comparison, we quantified the goodness of the obtained correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We find that the solar analogues selected for Gaia DR3 to compute the reflectance spectra of the asteroids of this data release have a systematically redder spectral slope at wavelengths shorter than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm than Hyades 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We find that no correction is needed in the red photometer (RP, between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='7 and 1 µm), but a correction should be applied at wavelengths below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm, that is in the blue photometer (BP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' After applying the correction, we find a better agreement between Gaia DR3 spectra, ECAS, HST, and our set of ground-based observations with the TNG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Correcting the near-UV part of the asteroid reflectance spectra is very important for proper comparisons with laboratory spectra (minerals, meteorite samples, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=') or to analyse quantitatively the UV absorption (which is particularly important to study hydration in primitive asteroids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' The spectral behaviour at wavelengths below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='5 µm of the selected solar analogues should be fully studied and taken into account for Gaia DR4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Gaia – asteroids – Solar analogues – UV – spectra 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Introduction Asteroid reflectance spectra and/or spectrophotometry pro- vide(s) information on their surfaces’ composition and the pro- cesses that modify their properties such as space weathering (Reddy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Historically, the use of photoelectric detec- tors (or photometers), which are more sensitive at bluer wave- lengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='5 µm), and the development of the standard UBV photometric system (Johnson & Morgan 1951) led to the appearance of the first asteroid taxonomies in the 1970s (Zell- ner 1973;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Chapman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 1975), which contained information at blue-visible wavelengths or what we call near-UV (NUV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' The introduction of the charge-coupled-devices (CCDs) in as- tronomy in the 1990s and later on contributed to the ’loss’ of NUV information, as CCDs were much less sensitive at those wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Therefore, the large majority of the modern spec- troscopic and spectrophotometric surveys cover the wavelength range from ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='5 µm up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Nevertheless, there are some exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' One of the first large surveys with information in the NUV is the Eight Asteroid Survey (ECAS, Zellner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' In this survey, we can find the photometry in eight broad-band filters between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='34 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='04 µm for 589 minor planets, includ- ing two filters below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='45 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' These observations were used to develop a new taxonomy (see Tholen 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Other recent cat- alogues, such as the Sloan Digital Sky Survey (SDSS) Moving Objects catalogue (Ivezi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2002), the Moving Objects Ob- served from Javalambre (MOOJa) catalogue from the J-PLUS survey (Morate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2021), or the Solar System objects obser- vations from the SkyMapper Southern Survey (Sergeyev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Article number, page 1 of 13 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='02157v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='EP] 5 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' main 2022), also include photometry in five, 12, and six filters be- tween 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='3 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='1 µm with 104,449, 3122, and 205,515 objects observed, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' The new Gaia data release 3 (DR3 here- after) catalogue, which was released in June 2022, offers 60,518 objects binned in 16 wavelengths between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='352 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='056 µm to mean reflectance spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Even though some laboratory measurements suggest the po- tential of the NUV absorption as a diagnostic region of hydrated and ferric material (Gaffey & McCord 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Feierberg 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Feierberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Hiroi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Cloutis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2011a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Hendrix et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Hiroi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2021), a quantitative distribu- tion of the NUV absorption among asteroids has not been dis- cussed before (Tatsumi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' The small sensitivity of CCDs and the lower Sun’s emission in NUV wavelengths make observations difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Moreover, the Rayleigh scattering by the atmosphere is stronger on shorter wavelengths, decreasing the signal-to-noise ratio (S/N) for the NUV region observed from the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' To compute the reflectance spectra, we needed to di- vide wavelength by wavelength of the measured spectra by the spectra of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' As it is unpractical to observe the Sun with the same instrument used to observe asteroids, we used solar analogues (SAs), that is stars selected by their known similar spectra to that of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' As the large majority of the spec- troscopic and spectrophotometric surveys cover the wavelength range that goes from the visible to near-infrared (NIR), the most commonly used SAs are well characterised at those wavelengths but they can behave very differently in the NUV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' This flux dif- ference at bluer wavelengths can introduce systematic errors in the asteroid reflectance spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' A good example is the work by de León et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' (2016), where they searched for the presence of F-type asteroids in the Polana collisional family since the parent body of the family, asteroid (142) Polana, was classified as an F type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' The authors obtained reflectance spectra in the NUV of the members of the family, finding that the large majority were classified as B types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' As most of the observers, they used SAs that were widely used by the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Interestingly, after ob- taining the asteroid reflectances again using only Hyades 64 as the SA, Tatsumi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' (2022) found that the large majority of the observed members of the Polana family were indeed F types and not B types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' This evidences the importance of using ade- quate SAs when observing in the NUV, and it has been the main motivation for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' In this Letter, we present a comparison between the SAs se- lected to compute the reflectance spectra in the frame of the data processing of Gaia DR3 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2022) and Hyades 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We analyse the results from this comparison and propose a multiplicative correction that can be applied to the archived asteroids’ reflectance spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We finally tested it by comparing corrected Gaia reflectance spectra with ground-based observations that have also been corrected against the same SA (ECAS survey, TNG spectra) and with one observation with the Hubble Space Telescope (HST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Sample 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Solar analogues in Gaia DR3 The Gaia DR3 catalogue (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2022) gives access to internally and externally calibrated mean spectra for a large subset of sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Internally calibrated spectra refer to an internal reference system that is homogeneous across all differ- ent instrumental configurations, while externally calibrated spec- tra are given in an absolute wavelength and flux scale (see De Angeli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Montegriffo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2022, for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Epoch spectra (spectra derived from a single observation rather than averaging many observations of the same source) are not included in this release.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' For this Letter, we relied on internally calibrated data when computing the correction for the Gaia re- flectances to ensure consistency and to avoid artefacts that could appear when dividing two externally calibrated spectra, as they are polynomial fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' To select the SAs, the Gaia team did a bibliographic search and selected a list of stars that are widely used as solar analogues for asteroid spectroscopy (Bus & Binzel 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Lazzaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Soubiran & Triaud 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Fornasier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Popescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Perna et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Popescu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Lucas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' First of all, we note that the star identified as 16 Cygnus B in Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' (2022) is in fact 16 Cygnus A and that the parameters in their Table C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' correspond to those of 16 Cygnus A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Luckily enough, the spectrum of 16 Cygnus B was also available in DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Among the referenced works, only Soubi- ran & Triaud (2004) carried out a search for SAs by comparing their spectra to that of the Sun down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='385 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' The rest sim- ply used G2V stars or cited previous works that presented SAs, as in Hardorp (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' In this later work, Hardorp selected SAs by comparing their spectra with the spectrum of the Sun using wavelengths down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='36 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' He highlighted the variations that can exist at NUV wavelengths even between stars of the same spectral class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Asteroids in Gaia DR3 Among the Gaia DR3 products for Solar System objects (SSOs), neither the internally nor the externally calibrated spectra are available to the community, as is the case for the stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' This is due to a specific choice of the Data Processing and Analysis Consor- tium (DPAC) caused by the difficulty of calculating those quanti- ties owing to the intrinsic variability and proper motion of SSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Instead, for each SSO and each epoch, the nominal, pre-launch dispersion function was used to convert pseudo-wavelengths to physical wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' The reflectance spectra were calculated by dividing each epoch spectrum by the mean of the SAs selected and then averaging over the set of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' After that, a set of fixed wavelengths every 44 nm in the range between 374 and 1034 nm was defined, with a set of bins centred at those wave- lengths and with a size of 44 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' For each bin (a total of 16 are provided), a σ-clipping filter was applied and a weighted aver- age using the inverse of the standard deviation as weight was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Finally, the reflectances were normalised to unity us- ing the value at 550±25 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' This final product is the only one available in DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Hyades 64 & 16 Cyg B As mentioned in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='1, Hardorp (1978) concluded that Hyades 64 and 16 Cyg B are two of the four stars that exhibit ’almost indistinguishable’ NUV spectra (quoting the author’s words) from the spectrum of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' This was confirmed in subsequent papers from the same author (Hardorp 1980a,b) and from other researchers (Cayrel de Strobel 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Porto de Mello & da Silva 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Farnham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Soubiran & Triaud 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We used these two stars as a ’reference’ to compute the correc- tion factor to be applied to the Gaia DR3 asteroids spectra, as they are in the list of SAs selected by Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' The methodology is described in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We note that the obtained correction factor using Hyades 64 as opposed to 16 Cygnus B differs less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We, therefore, Article number, page 2 of 13 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Tinaut-Ruano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' : Asteroids’ reflectance from Gaia DR3: Artificial reddening at near-UV wavelengths Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Multiplicative correction factors for Gaia asteroid binned spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We include the wavelengths below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Wavelength (µm) Correction factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='374 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='418 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='462 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='506 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='550 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='00 decided to use Hyades 64, as it was the star that was used for both the ECAS survey and our ground-based observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Computing the correction factor: Internally calibrated data In order to compute a correction applicable to the Gaia DR3 reflectances, we proceeded as follows: first, using the internally calibrated data, we computed the ratio between the Gaia sample of SAs, as well as the mean spectrum of these SAs, and Hyades 64 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' As we can observe in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 1, which corresponds to the red photometer (RP), the deviation from the unity of the ratio between Gaia’s mean SA and Hyades 64 (black line) is always below 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Therefore, this mean spectrum can confidently be used to obtain the reflectance spectra of asteroids above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' However, the situation in the the blue photometer (BP) is quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We can see in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 1 that the de- viation from the unity of the above defined ratio can reach values of up to 10%, indicating that the mean spectrum of the SAs used in Gaia DR3 differs significantly from Hyades 64 at wavelengths below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' The biggest effect when using this mean spec- trum to obtain asteroids’ reflectance spectra is the introduction of a systematic (and not real) positive slope, in particular in the range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='4 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm, mimicking a drop in reflectance below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Furthermore, the division by this mean spec- trum can also introduce a ’fake’ absorption around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='38 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We have quantified this spectral slope in two separate wavelength ranges, trying to reproduce the observed behaviour of the ratio: one slope between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='4 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm, which we named S Blue, and another one for wavelengths below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='4 µm, named µm S UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' The obtained values for the individual SAs used in Gaia DR3 (blue stars), as well as for the mean spectrum (blue cross) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' For the mean spectrum of the SAs used in Gaia DR3, we found that the introduced slopes are S Blue = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='38 µm−1 and S UV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='69 µm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' From this analysis, we conclude that a correction is needed in the NUV wavelengths, that is below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' To arrive at the multiplicative correction factors, we binned the ratio between the mean spectra of SAs selected by the DPAC and Hyades 64, using the same wavelengths and bin size as the ones adopted for the asteroid reflectance spectra in the Gaia DR3 (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' In this way, the users can easily correct the asteroid spectra at NUV wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' The obtained values are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Comparison of corrected reflectances with existing data To correct the artificial slopes introduced by the use of the mean Gaia SAs, we multiplied the binned asteroid reflectance spec- tra below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm by the corresponding correction factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We compared the corrected Gaia spectra with spectra or spectropho- tometry of the same asteroids obtained using other facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' As a first step, we selected only those Gaia asteroid spectra with a S/N > 160, as we detected a systematic decrease in spectral slope values at blue wavelengths with decreasing S/N for objects with a smaller S/N than 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We then selected spectrophotomet- ric data from the ECAS survey for asteroids that have more than one observation, and NUV spectra obtained with the Telesco- pio Nazionale Galileo (TNG) and previously published by Tat- sumi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' The resulting comparison dataset is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='1, where the red lines correspond to the original Gaia re- flectances, black lines are the corrected ones, dark blue lines cor- respond to ECAS data, and TNG spectra are shown in light blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' As can be seen, the corrected reflectances are in better agree- ment with the ECAS and TNG data than the original ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We also included the UV spectrum of asteroid (624) Hector down- loaded from the ESA archive using the python package astro- query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='hubble1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' It was obtained with STIS at HST (Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We converted the flux to reflectance using the spec- trum of the Sun provided for the STIS instrument2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We note that even after the correction, some asteroids show discrepancies with the reference data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' This is discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Results and discussion We have shown that the artificial slope introduced at blue wavelengths in the Gaia DR3 asteroid data due to the selected SAs is -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='38 µm−1 in the range between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='4 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='69 µm−1 below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='4 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Following Zellner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' (1985), the b and v filters of the ECAS survey have central effective wavelengths of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='437 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='550 µm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' According to Tholen (1984), the (b-v) colours of the mean F and B taxonomical classes are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='049 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='015 magnitudes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Transforming these colours to relative reflectances results in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='046 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='014, which gives slopes of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='407 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='124 µm−1 between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='437 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Therefore, the difference between these computed slopes for F and B types (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='283 µm−1) is smaller than the artificial slope introduced by the use of the mean SA of Gaia, implying that unless we apply the correction proposed in this Letter, asteroids can be easily misclassified as B types when actually being F types (see the described example in the Introduction for the case of members of the Polana family).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' To test and quantify the goodness of our proposed correction, we computed the spectral slope between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='437 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm for the ECAS comparison dataset, and between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='418 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 for Gaia original and corrected spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 3 we plotted the difference between those slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' After applying our correction factor, we could see that the large majority (148 out of 152) of the asteroids have more similar slopes to those of ECAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Nevertheless, our correction has limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' First, we were testing its goodness over space-based observations using ground-based observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' For wavelengths down to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='3 µm, ground-based observations present some difficulties, mainly due to the atmospheric absorption and the lower sensitivity of the detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Furthermore, Gaia observations at those wavelengths also have other artifacts that we do not fully understand, such as the detected strong decrease in the spectral slope below S/N 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Another point to consider when comparing asteroid spectra observed in different epochs is the effect of the different viewing geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' This difference in the viewing geometry, and thus, in 1 https://astroquery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='io/en/latest/esa/ hubble/hubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='html 2 https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='edu/hlsps/reference-atlases/ cdbs/current_calspec/sun_reference_stis_002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='fits Article number, page 3 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' main 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='60 Wavelength [ m] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='15 Counts relative to Hyades 64 BP 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='0 Wavelength [ m] RP HD060234 HD123758 16 Cyg B(A)1 HD6400 HD220022 HD220764 HD016640 HD292561 HD100044 HD155415 SA110-361 HD182081 HD144585 HD146233 HD138159 HD139287 HD020926 HD154424 HD202282 16 Cyg B mean Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Ratio between the internally calibrated spectra of each of the Gaia SAs and Hyades 64 in the blue photometer (BP, left panel) and the red photometer (RP, right panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We also plotted the ratio of the mean Gaia SA and Hyades 64 (black solid line) and the binned version of this ratio at the wavelengths provided for SSO in Gaia DR3 (black dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 1 We note that the star identified as 16 Cygnus B in Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' (2022) is in fact 16 Cyg A (see the main text for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='5 SUV [ m 1] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='0 SBlue [ m 1] Gaia SAs Gaia mean Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Slopes introduced by each of the SAs in the Gaia sample (blue stars) and their mean (blue cross), compared to Hyades 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We note that S Blue was computed in the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='4–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm range, while S UV was computed using wavelengths below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='4 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' the phase angle, causes a change in the spectral slope known as phase reddening or phase coloring Alvarez-Candal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' This effect has not been well studied at blue wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Still, even in the event that we were able to correct it, Gaia’s spectra are, on average, over different epochs and the information on the phase angle values is not provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='0 ECAS slope - original Gaia slope (1/ ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='0 ECAS slope - corrected Gaia slope (1/ ) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Difference between the blue slope for ECAS and for Gaia origi- nal data (x-axis) and corrected data (y-axis) in the comparison sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Conclusions We have found that the use of the SAs selected to compute the reflectance spectra of the asteroids in Gaia DR3 introduces an artificial reddening in the spectral slope below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='5 µm, that is an artificial drop in reflectance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' By comparing those SAs with Hyades 64, one of the best characterised SAs at NUV wave- lengths, we obtain multiplicative correction factors for each of the reflectance wavelengths below 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='55 µm (a total of four) that can be applied to the asteroids’ reflectance spectra in Gaia DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' By applying this correction, we found a better agreement be- tween the Gaia spectra and other data sources such as ECAS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Article number, page 4 of 13 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Tinaut-Ruano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' : Asteroids’ reflectance from Gaia DR3: Artificial reddening at near-UV wavelengths The behaviour of the SAs in the red wavelengths is in agree- ment with Hyades 64 within 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' This was somehow expected, as the majority of the SAs used by the Gaia team were previ- ously tested and widely used by the community to obtain visible reflectance spectra of asteroids, typically beyond 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='45–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Correcting the NUV part of the asteroid reflectance spectra is fundamental to study the presence of the UV absorption, which has been associated with hydration in primitive asteroids, or to discriminate between B and F types, which are two taxonom- ical classes that have proven to have very distinct polarimetric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' The NUV region has not yet been fully exploited for asteroids and, in this way, Gaia spectra constitute a major step forward in our understanding of these wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' FTR, JdL, ET, DM, and JL acknowledge support from the Agencia Estatal de Investigación del Ministerio de Ciencia e Innovación (AEI- MCINN) under the grant ’Hydrated Minerals and Organic Compounds in Prim- itive Asteroids’ with reference PID2020-120464GB-100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' FTR also acknowledges the support from the COST Action and the ESA Archival Visitor Programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' DM acknowledges support from the ESA P3NEOI programme (AO/1- 9591/18/D/MRP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='int/web/gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Funding for the DPAC has been pro- vided by national institutions, in particular, the institutions participating in the Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' The work of MD is supported by the CNES and by the project Origins of the French National Research Agency (ANR-18-CE31-0014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' De Angeli is supported by the United Kingdom Space Agency (UKSA) through the grants ST/X00158X/1 and ST/W002469/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' References Alvarez-Candal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=', Jimenez Corral, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=', & Colazo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 2022, A&A, 667, A81 Bus, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' & Binzel, R.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=', Tholen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=', & Tedesco, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' 1985, Icarus, 61, 355 Article number, page 5 of 13 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' main Appendix A: Comparison figures In this appendix, we are showing the spectra of asteroids that have at least two observations from the ground from ECAS (dark blue) or from TNG (light blue) and also have available spectra in Gaia DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' We plotted the original (red) and the corrected (black) version of the Gaia spectra together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' For asteroid 624 (Hector), we also added an observation from HST (further information can be found in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Article number, page 6 of 13 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' Tinaut-Ruano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content=' : Asteroids’ reflectance from Gaia DR3: Artificial reddening at near-UV wavelengths 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='8 1 ECAS original Gaia corrected Gaia 2 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='8 4 6 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tA0T4oBgHgl3EQfOP83/content/2301.02157v1.pdf'} +page_content='0 0.' metadata={'source': 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It allows for large-scale parallelism that leads to significantly reduced +computational time compared to serial time-stepping methods. However, like many parallel-in- +time methods it can fail to achieve parallel speedup when applied to non-diffusive equations +such as hyperbolic systems or dispersive nonlinear wave equations. This paper explores the use +of exponential integrators within the Parareal iteration. Exponential integrators are particularly +interesting candidates for Parareal because of their ability to resolve fast-moving waves, even at +the large stepsizes used by coarse propagators. This work begins with an introduction to expo- +nential Parareal integrators followed by several motivating numerical experiments involving the +nonlinear Schrödinger equation. These experiments are then analyzed using linear analysis that +approximates the stability and convergence properties of the exponential Parareal iteration on +nonlinear problems. The paper concludes with two additional numerical experiments involving +the dispersive Kadomtsev-Petviashvili equation and the hyperbolic Vlasov-Poisson equation. +These experiments demonstrate that exponential Parareal methods can achieve significant +parallel speedup on different types of non-diffusive equations. +1. Introduction +Time integrators [37, 72, 12] are numerical methods that solve an initial value problem by sequentially advancing +the solution via a series of discrete timesteps. For more than half a century, these methods have proven invaluable +for modeling a range of dynamical processes appearing in both science and engineering. In a typical calculation one +iteratively applies a time integrator to evolve a system over thousands or even millions of timesteps. Therefore, the total +computational cost is not just that of a single timestep, but rather the combined cost of applying the method sequentially +over the full temporal domain. +The sequential nature of classical time-stepping methods has come under increasing scrutiny in light of modern +parallel hardware like multicore processors, massively parallel high performance computing systems, and specialized +accelerators. For more than two decades, these advancements have spurred the development of new parallel-in-time +(PinT) methods [44, 54, 24, 27, 30] that distribute the full temporal domain over a large number of computational nodes. +Perhaps the most well-known PinT method is the Parareal algorithm [54]. Parareal consists of a parallel iteration +that combines a fine propagator (a computationally expensive and accurate integrator) with a coarse propagator (a +computationally cheap and less accurate integrator). The aim of Parareal is to obtain the solution of the fine propagator +at a similar computational cost to that of the coarse propagator. +Parareal has proven effective for accelerating the solution of diffusive equations [26, 71, 58, 52] and its theoretical +convergence properties are well understood in the presence of diffusion [33]. In contrast, non-diffusive equations (e.g. +hyperbolic systems or dispersive nonlinear wave equations) introduce significant numerical difficulties that lead to +slow convergence or instabilities in the Parareal iteration [8, 69, 31, 64, 15]. Though numerous modifications have +been proposed [19, 22, 23, 25, 32, 51], the resulting methods introduce additional complexities that make them less +applicable to all types of problems. +In this work we combine exponential integrators [43] with Parareal and demonstrate, both theoretically and +experimentally, that this pairing can provide parallel speedup on non-diffusive equations. We focus specifically on +⋆This work was funded by the National Science Foundation, Computational Mathematics Program DMS-2012875 +∗Corresponding author +tbuvoli@tulane.edu (T. Buvoli); mlminion@lbl.gov (M. Minion) +ORCID(s): +T Buvoli et al.: Preprint submitted to Elsevier +Page 1 of 33 +arXiv:2301.03764v1 [math.NA] 10 Jan 2023 + +aExponential Runge-Kutta Parareal +the non-diffusive, semilinear initial value problem +퐲′ = 퐋퐲 + 푁(푡, 퐲), +퐲(푡0) = 퐲0 +(1) +where the eigenvalues of both 퐋 and 휕푁 +휕퐲 (the Jacobian of 푁(푡, 퐲)) are purely imaginary. We were motivated to consider +exponential integrators because they treat the linear component 퐋 exactly, granting them the ability to accurately resolve +fast moving waves even at coarse stepsizes. This property is beneficial because the convergence of Parareal on non- +diffusive problems depends critically on accuracy differences between the coarse and fine solver [64, 15]. However, +special care must be taken when applying exponential integrators on non-diffusive equations since the methods are +classically unstable [17, 21]. In fact, we will demonstrate that the repartitioning strategy introduced in [17] is essential +for obtaining stable exponential Parareal methods on stiff non-diffusive problems. +The organization of this paper is as follows. In section 2 and section 3 we respectively introduce Parareal +and exponential integrators. In section 4 we motivate this paper by presenting several numerical experiments +involving the one-dimensional nonlinear Schrödinger equation. Then, in section 5 we introduce analytical tools for +understanding the convergence and stability properties of a Parareal configuration. Lastly, in section 6 we present +additional numerical experiments that demonstrate exponential Parareal’s ability to achieve parallel speedup on higher- +dimensional hyperbolic and dispersive wave equations. +2. Parareal Introduction +In this section we describe the Parareal algorithm [54], present a formula for parallel speedup, and provide a +complete table of all Parareal parameters. We begin by supposing that one seeks an accurate, numerical solution to the +initial value problem +퐲′(푡) = 푓(퐲(푡)), +퐲(푡0) = 퐲0. +(2) +If computational cost can be neglected, then an accurate numerical integrator , such as a high-order method with +small timesteps, should be considered. However, this will not always be practical since the time to run the calculation +can become prohibitive. Therefore, we often settle for a less accurate integrator , such as a low-order method that is +run using larger timesteps. Can the situation be improved with access to parallel computational hardware? +The Parareal method is a parallel iteration that combines a coarse propagator  with a fine propagator , and +converges to the solution of . Provided that the iteration can be efficiently parallelized and that the convergence +rate is sufficiently high, then the computational time needed to run Parareal is similar to that of running the coarse +propagator. In the following subsections, we explore the algorithm in more detail. +2.1. Method definition +Let  and  be two one-step methods that are respectively called the coarse and fine propagators; it is assumed +that  is more computationally expensive to apply than . Next, suppose that we want to approximate (2) at a discrete +set of time points using the fine propagator, such that +푦푛+1 = (푦푛), +푛 = 0, … , 푁푝, +(3) +where 푦푛 ≈ 푦(푡푛). The Parareal algorithm converges to (3) by taking a provisional solution, 푦0 +푛 ≈ 푦(푡푛), that is usually +computed by a serial application of the coarse propagator , and then correcting it via the iteration +푦푘+1 +푛+1 = (푦푘+1 +푛 +) + (푦푘 +푛) − (푦푘 +푛), +{ +푛 = 0, … , 푁푝, +푘 = 0, … , 퐾 − 1. +(4) +In order to run the Parareal iteration, it is necessary to store and iteratively update the solution values along the entire +time interval. The key property of the Parareal iteration is that the fine integrator  can be applied in parallel on 푁푝 +processors. To further clarify this point, we show pseudocode for the Parareal iteration (4) in table 1. +T Buvoli et al.: Preprint submitted to Elsevier +Page 2 of 33 + +Exponential Runge-Kutta Parareal +Parareal Pseudocode +1. +% provisional solution +2. +for n = 0 : 푁푝 − 1 +3. +푦0 +푛+1 = (푦0 +푛) +4. +% Parareal iteration +5. +for k = 0 : K - 1 +6. +parfor j = 0 : 푁푝 − 1 % parallel loop +7. +퐹푗 = (푦푘 +푗 ) +8. +for j = 0 : 푁푝 − 1 +9. +푦푘+1 +푗+1 = (푦푘+1 +푗 +) + 퐹푗 − (푦푘 +푗 ) +10. +return 푦퐾 +푁푝 +Table 1 +Pseudocode for the Parareal iteration (4). The fine integrator (colored in red) can run in a parallel loop, while the loops +containing the coarse propagator (colored in blue) are serial. Pseudocode for more efficient pipelined implementations are +contained in [7, 62]. +2.2. Parallel speedup +Parallel speedup is defined as the ratio between the computational time for running a serial algorithm and its parallel +equivalent. For Parareal we compute speedup by dividing the computational cost of the sequential fine integrator (3) +by the computational cost of the Parareal iteration (4) when run using 푁푝 processors. Let the cost for a single step +of the fine propagator  and the coarse propagator  be 퐶 and 퐶, respectively. The cost of performing 퐾 Parareal +iterations is the sum of the cost of the predictor, 푁푝퐶, plus the additional cost of each iteration which, neglecting +communication, is 퐾(퐶 + 퐶); see [7]. In summary, the serial cost for computing (3) is 퐶푠 = 푁푝퐶 and a parallel +cost for (4) is 퐶푝 = 푁푝퐶 + 퐾(퐶 + 퐶). If we let 훼 = 퐶∕퐶, then the parallel speedup is +푆 = 퐶푠 +퐶푝 += +푁푝 +푁푝훼 + 퐾(1 + 훼). +(5) +Lastly we remark the speedup formula will change if one considers more elaborate parallelization strategies such as +those presented in [7, 9, 4, 63]. +2.3. Selecting the coarse and fine propagators +A user can select any pair of one-step methods to be the coarse and fine propagators. A common approach, which +we will use in this work, is to set the coarse propagator  equal to 푁푔 steps of an inexpensive one-step method 푔 and +the fine propagator  equal to 푁푓 steps of an expensive integrator 푓. Both  and  must advance the solution by the +same amount; therefore, if we let ℎ be the stepsize of 푓, then the stepsize of 푔 must be ℎ푁푓∕푁푔. If we use the notation +푀휅(휂) to denote 휅 steps of a the method 푀 run with stepsize 휂, then the fine and coarse propagators are + = 푓 푁푓 (ℎ) +and + = 푔푁푔(ℎ푁푓∕푁푔). +(6) +In fig. 1 we illustrate the the resulting coarse and fine grids for the coarse and fine propagators (6). Since the fine +propagator  is now 푁푓 steps of the method 푓, a Parareal method that converges to the solution of  applied over 푁푝 +steps is also converging to the solution of 푓 applied over total of 푁푠 = 푁푓푁푝 steps. Throughout this work we will +frequently characterize Parareal in terms of (푓, 푁푓), (푔, 푁푔), and 푁푠 instead of , , and 푁푝. +2.4. A complete table of parameters +As we have seen, the Parareal algorithm has a large number of free parameters. In table 2 we make a complete +list of parameters that are relevant to this work. As we note in the table, the integer variables 푁푝, 푁푠, 푁푏 and 푁푓 are +related by the equation 푁푠 = 푁푏푁푝푁푓 and therefore the user can only select three of these variables with the fourth +being automatically determined. +T Buvoli et al.: Preprint submitted to Elsevier +Page 3 of 33 + +Exponential Runge-Kutta Parareal +G +f(h) +f(h) +f(h) +g(3h) +F +Figure 1: An illustration of the coarse and fine propagators (blue arrow for coarse and red arrows for fine), the coarse +temporal grid (large, black squares), and fine temporal grid (small, white circles). This illustration depicts the following +parameters: the time interval has been divided into 12 total timesteps (푁푠 = 12), the coarse propagator  that takes a +single step of 푔 (푁푔 = 1), the fine propagator  that takes three steps of 푓 (푁푓 = 3), and the resulting Parareal iteration +requires 4 processors (푁푝 = 4). +Interdependent User Defined Parameters +(A user must select two in such a way that all three variables are integers) +Variable +Meaning +푁푝 +Number of processors +푁푠 +Number of fine steps +푁푓 +Number of RK steps in +Interdependency +푁푠 = 푁푝푁푓 +Independent User Defined Parameters (A user must select all) +Variable +Meaning +푓 +RK method used in  +푔 +RK method used in  +푁푔 +Number of RK steps in  +퐾 +Number of Parareal iterations +Dependent Parameters +Variable +Meaning +Definition +ℎ +Time step for serial method +푡final∕푁푠 + +Coarse propagator +푁푔 steps of RK method 푔 + +Fine propagator +푁푓 steps of RK method 푓 +푁푇 +Total number of fine steps per block +푁푠∕푁푏 +푐푔 +Cost of method per step in  +User defined +푐푓 +Cost of method per step in  +User defined +퐶 +Cost of  +푁푓푐푓 +퐶 +Cost of  +푁푔푐푔 +Table 2 +Parareal parameters names and definitions that are relevant to this work. +3. Exponential integrators +The aim of this work is to study Parareal methods where the coarse and fine propagators are exponential Runge- +Kutta methods. In this section we provide an introduction to exponential integrators and discuss their stability properties +on non-diffusive equations. Exponential integrators [43] are a class of numerical methods for solving the semilinear +initial value problem +퐲′ = 퐋퐲 + 푁(푡, 퐲), +퐲(푡0) = 퐲0. +(7) +In the past two decades, they have proven highly efficient for solving stiff systems and can offer certain advantages over +both fully-implicit and linearly-implicit methods [36, 48, 55, 57, 40, 41]. The main idea behind exponential integrators +T Buvoli et al.: Preprint submitted to Elsevier +Page 4 of 33 + +Exponential Runge-Kutta Parareal +is to consider the exact solution to (7), namely +퐲(푡0 + ℎ) = 푒ℎ퐋퐲0 + ∫ +푡0+ℎ +푡0 +푒(푡0+ℎ−휏)퐋푁(휏, 퐲(휏))푑휏, +(8) +and replace the nonlinear term 푁(휏, 퐲(휏)) with an explicit polynomial approximation. Different approximations lead +to different families of exponential integrators, including exponential linear multistep methods [10], Runge-Kutta +methods [20, 53, 42, 56, 13], and general linear methods [61, 14]. +Since 푁(휏, 푦(휏)) is approximated with a polynomial, the coefficients of all exponential integrators depend on a +subset of the exponential functions +휑0(ℎ퐋) = 푒ℎ퐋 +and +휑푗(ℎ퐋) = ∫ +1 +0 +푒(1−푠)ℎ퐋푠푗푑푠 +(푗 ≥ 1). +(9) +At each timestep an exponential integrator requires matrix-vector products with the 휑-functions of the linear operator +퐋. For many problems this can be done efficiently using a number of different algorithms [6, 18, 39], including those +based on squaring methods [50, 3, 2], contour integration [48, 70], Krylov-subspaces [40, 41, 59, 60, 34], and parallel +rational approximations [38, 66, 65]. +3.1. Exponential Runge-Kutta methods +Exponential Runge-Kutta (ERK) methods are one-step methods that approximate the solution to (2) by taking a +linear combination of stage values at each timestep . The simplest ERK integrator is the exponential Euler method that +is obtained by replacing 푁(휏, 퐲(휏)) in (8) with the constant approximation 푁(푡푛, 퐲푛), yielding +퐲푛+1 = 휑0(ℎ퐋)퐲푛 + 휑1(ℎ퐋)ℎ푁(푡푛, 퐲푛). +(10) +More generally, an 푠-stage ERK method is +푌푖 = 휑0(ℎ푐푗퐋)퐲푛 + +푖−1 +∑ +푗=1 +푎푖푗(ℎ퐋)푁(푐푗, 푌푗), +푖 = 1, … , 푠, +(11) +퐲푛+1 = 휑0(ℎ퐋)퐲푛 + +푠 +∑ +푗=1 +푏푗(ℎ퐋)푁(푐푗, 푌푗) +(12) +where 푎푖푗(ℎ퐋) and 푏푗(ℎ퐋) are functions that include linear combinations or products of the 휑-functions (9). By applying +the identity 휑0(ℎ퐋)퐲푛 = 퐲푛+휑1(ℎ퐋)퐋퐲푛 one can rewrite the equations (11) and (12) in terms of 휑푗(ℎ퐾) for 푗 ≥ 1; this +can be advantageous both for method analysis and implementation. Lastly, like classical RK methods, ERK methods +can be represented using the Butcher Tableau +푐1 +0 +푐2 +푎21 +0 +⋮ +⋮ +⋱ +⋱ +푐푠 +푎푠,1 +… +푎푠,푠−1 +0 +푏1 +… +푏푠−1 +푏푠 +where the coefficients 푎푖푗 and 푏푗 are now matrix functions of the linear operator ℎ퐋. +In this work, we will consider ERK methods of orders one to four from [20, 53]. We name these methods ERK1, +ERK2, ERK3, and EKR4, and list their Tableaus in appendix A. +3.2. Stability and repartitioning for non-diffusive equations +Since exponential integrators treat the linear operator 퐋 exactly, we would expect that they offer significantly +improved stability properties compared to explicit integrators. While this is true for diffusive operators, the situation +is more nuanced when 퐋 has purely imaginary eigenvalues [17, 21]. In particular, both exponential and explicit +integrators have similarly sized stability regions, but the magnitude of the instabilities is often very small for exponential +T Buvoli et al.: Preprint submitted to Elsevier +Page 5 of 33 + +Exponential Runge-Kutta Parareal +integrators. Therefore, unlike explicit methods, exponential integrators can still produce usable solutions on stiff non- +diffusive equations so long as the total number of timesteps is not overly large [17]. +In [17] we proposed a strategy that stabilizes exponential integrators by repartitioning the right-hand-side of +(7) using perturbed linear and nonlinear operators ̂퐋 and ̂ +푁. This enables long-time simulations with exponential +integrators and also removes instabilities when the underlying equation focuses energy into unstable modes. The +perturbed operators are formed by respectively adding and subtracting a diffusive operator 퐃 such that +̂퐋 = 퐋 + 휖퐃 +and +̂ +푁(푡, 퐲) = 푁(푡, 퐲) − 휖퐃. +(13) +In short, we add damping to the linear operator ̂퐋 and excitation to the nonlinear operator ̂ +푁(푡, 퐲). The differential +equation (7) can then be written in terms of the perturbed operators as +퐲′ = ̂퐋퐲 + ̂ +푁(푡, 퐲). +(14) +Therefore, an exponential integrator that solves the repartitioned equation (14) is simultaneously solving (7). However, +instead of treating 퐋 exactly and approximating 푁(푡, 퐲), a repartitioned integrator treats ̂퐋 exactly and approximates +̂ +푁(푡, 퐲). The advantage of repartitioning is that the exponential integrator now possesses a large stability region for a +continuous range of small 휖 values [17]. If 퐋 is diagonalizable, such that 퐋 = 퐔횲퐔−1, and we select +퐃 = −퐔|횲|퐔−1 +and +휖 = +1 +tan(휋∕2 − 휌) +for +휌 ∈ (0, 휋∕2), +(15) +then we rotate all the eigenvalues of a non-diffusive linear operator 퐋 by 휌 degrees into the left-half plane. In other +words, the eigenvalues of ̂퐋 all lie on the wedge 푟푒푖휃 for 푟 ≥ 0 and 휃 ∈ {휋∕2 + 휌, 3휋∕2 − 휌}. Many other choices for +퐃 are possible (e.g. low-order, even spatial derivatives), however in [17] we proposed (15) because of its convenience +when analyzing the stability effects of repartitioning. +In summary, exponential integrators exhibit mild instabilities for stiff non-diffusive equations that can be eliminated +through re-partitioning. In sections 4 and 5 we will show that these instabilities are greatly exacerbated by the Parareal +iteration, and that repartitioning is essential for obtaining stable, exponential Parareal methods for solving stiff non- +diffusive equations. +4. Motivating numerical experiments +In this section we present three numerical experiments that highlight key properties of exponential Parareal +integrators applied to non-diffusive equations. Later, in section 5 we will see how linear stability analysis and linear +convergence analysis can be used to more rigorously quantify our results. All three numerical experiments involve the +one-dimensional nonlinear Schrödinger (NLS) equation +푖푢푡 + 푢푥푥 + 2|푢|2푢 = 0 +(16) +on the domain 푥 ∈ [−4휋, 4휋] with periodic boundary conditions. We discretize the equation in space using a 1024 +point Fourier spectral method that is dealiased using the classical 3∕2 rule. The equation is then integrated in Fourier +space where the derivative operators are diagonal. This results in the semilinear equation (7) with +퐋 = diag(−푖퐤2) +and +푁(푡, 퐲) = 2푖(−1(퐲) .* abs(−1(퐲))), +(17) +where 퐤 is a vector of Fourier wavenumbers and  denotes the discrete Fourier transform. To ensure the classical +stability of exponential integrators we also consider the repartitioning (13) and (14) where the diffusive operator 퐃 and +휖 are selected according to (15) such that +퐃 = diag(−퐤2), +휖 = +1 +tan(휋∕2 − 휌) +and +휌 = +휋 +128. +(18) +In addition to investigating stability and convergence, we also compare the theoretical speedup of the Parareal +iteration to its real-world performance on a distributed memory system. To do this, we implemented the exponential +T Buvoli et al.: Preprint submitted to Elsevier +Page 6 of 33 + +Exponential Runge-Kutta Parareal +푁푝 +2048 +푁푓 +32 +푓 +ERK4 +퐾 +1, … , 6 +푁푠 +216 +푁푔 +1 +푔 +ERK3 +Table 3 +Parareal parameters used for the nonlinear Schrodinger equation (16) with initial conditions (19) and (20). +Parareal method as part of the open source package LibPFASST1 and performed the numerical experiment on the Cray +XC40 Cori at the National Energy Research Scientific Computing Center. +Our first experiment uses the initial condition +푢(푥, 푡 = 0) = 1 + +1 +100 cos(푥∕4), +(19) +integrated out to time 푡 = 14. Repartitioning is not required for serial ERK methods on this short time-scale, and +the convergence curves for classical and repartitioned exponential integrators look identical (see convergence plots +in appendix B). We now consider two Parareal methods: one with classical ERK integrators, and the other with +repartitioned ERK integrators. To obtain a high-accuracy solution, we select ERK4 as the fine integrator 푓, and +푁푠 = 216 as the total number of fine steps; from the serial ERK convergence diagrams in fig. 17 we see that a fully +converged Parareal method will yield the solution with an error of 3 × 10−9. Next we must select a coarse integrator 푔 +that is stable at large stepsizes. Though it may seem tempting to select ERK1 because it is the least expensive method +per timestep, its poor stability leads us to choose the more stable ERK3 method. Lastly, we select 푁푔 = 1 and 푁푓 = 32; +this implies that the coarse propagator  consists of a single step of 푔, while the fine propagator  consists of 32 steps +of 푓. With these parameters the serial coarse propagator has an accuracy of 2 × 10−1, which is approximately eight +orders of magnitude less than the serial fine integrator (see the black crosses in fig. 17). A complete list of the Parareal +parameters used for this experiment is contained in table 3. +The computation was distributed on sixty-four, 32-core Intel Haswell nodes that provided a total of 2048 compute +cores. In fig. 2 we show how the error of the solution obtained by the Parareal iteration evolves as a function of the +iteration number. We also show the corresponding theoretical and achieved parallel speedup, along with a space-time +plot of the NLS solution. The results demonstrate that repartitioning is essential for obtaining a convergent Parareal +iteration; moreover, by using repartitioned ERK methods, Parareal is able to obtain a high-accuracy solution up to 38.5 +times faster than a serial ERK4 method. In contrast unmodified exponential integrators lead to a divergent Parareal +iteration whose error increases monotonically for iteration number 푘 greater than two. +Our timing results also reveal a practical challenge that can occur when applying a PinT method on a distributed +memory system. Using the Parareal configuration from table 3, we were only able to achieve a speedup factor of 10.4; +approximately one quarter less than the theoretical speedup factor of 38.5 predicted by (5). This difference is due to +unaccounted communication overhead. In fact, (5) will only be accurate if the time required to compute a single step of +the propagator  is significantly greater than the time for transferring the solution vector between two nodes. Although +this does not hold true for the one-dimensional NLS equation, the exponential Parareal method is neverthless able to +provide a high-accuracy solution an order of magnitude faster than the serial ERK4 method. Moreover, in section 6 +we will see that theoretical speedup very accurately predicts achievable speedup on more computationally expensive +two-dimensional problems. +For our second experiment we consider a modified initial condition that contains a high-frequency component, +namely +푢(푥, 푡 = 0) = 1 + +1 +100 +[cos(푥∕4) + cos(45푥∕4)] . +(20) +The newly added perturbation does not fundamentally change the solution, but rather introduces a low-amplitude +oscillation that persists throughout the temporal integration window; see fig. 3(b)-(c). Moreover, the high-frequency +mode does not make the computation more challenging for serial ERK methods, as evidenced by the convergence and +efficiency plots that are nearly identical to those generated using the initial condition (19); compare figs. 17 and 18. We +again consider the Parareal method from table 3 with either classical or repartitioned ERK integrators. In fig. 3(a) we +show error as a function of the iteration number 푘, and see that repartitioning is again required to prevent instabilities. +However, the high-frequency component has now prevented the repartitioned Parareal method from fully converging; +1https://github.com/libpfasst/LibPFASST +T Buvoli et al.: Preprint submitted to Elsevier +Page 7 of 33 + +Exponential Runge-Kutta Parareal +(a) Error versus Iteration +(b) Parallel Speedup versus Iteration +(c) NLS Solution +Figure 2: Error and speedup of the Parareal configuration from table 3 applied to the NLS equation (16) with the smooth +initial condition (19). Subfigure (a) shows the error at 푡final = 14 as a function of the Parareal iteration 푘. Line color is +used to distinguish classical exponential integrators from repartitioned exponential integrators. When 푘 = 0 the Parareal +method is equivalent to running the coarse integrator with 푁푠∕푁푓 steps. Subfigure (b) shows the parallel speedup as a +function of the iteration 푘; this compares the running time of the Parareal iteration to that of taking 푁푠 serial steps with +the fine integrator 푓. Note that speedup is identical for both classical and repartitioned Parareal. Lastly, figure 2(c) shows +the magnitude squared NLS solution |푢(푥, 푡)|2. +(a) Error versus Iteration +(b) Solution at 푡 = 14 +(c) Magnified Solution at 푡 = 14 +initial condition is (19) +initial condition is (20) +Figure 3: Subfigure (a) shows the error of the Parareal configuration from table 3 applied to the NLS equation (16) with +the oscillatory initial condition (20). Subfigure (b) compares the NLS solution at time 푡 = 14 for the two initial conditions +(19) and (20) that are respectively drawn using a thick gray line and thin black line. The region enclosed by a blue square +is magnified in subfigure (c) to highlight the small amplitude oscillation that arises from the initial condition (20). +the method remains stable as 푘 increases, however the error does not improve beyond 3×10−7. This is our first indication +that highly-oscillatory solutions will cause convergence problems for Parareal. +Thus far, we have seen that repartitioning is essential for preventing instabilities in the exponential Parareal iteration. +Therefore, we will no longer consider Parareal with classically partitioned ERK methods in this section. However, we +have also seen that repartitioning does not guarantee convergence. In [15, 64] it was shown that Parareal convergence +on non-diffusive problems improves when the coarse propagator  more closely approximates the fine propagator . +In our final motivating experiment, we will explore this phenomenon using an even more challenging initial condition +that contains 45 spatial modes, namely +푢(푥, 푡 = 0) = 1 + +1 +100 +45 +∑ +푘=1 +cos(푘푥∕4). +(21) +The NLS solution is now full of high-frequency information (see fig. 20) that causes even serial ERK integrators to +achieve slightly diminished accuracy for the same number of steps; compare fig. 19 to fig. 18. Based on the previous +experiment we expect that the high-frequency oscillations will prevent the Parareal configuration in table 3 from rapidly +T Buvoli et al.: Preprint submitted to Elsevier +Page 8 of 33 + +T +714010510X-1003 +Iteratio4 +n k56Fine + Errort +a +10 +110 +4 +1 +1Iclassicalrepartitioned10-10250 +40 +oeedup +30theoreticalachievedS +20 +10 +0 +12 +3 +4 +Iteration k5 +63 +Iteratio4 +n k56Fine +Errort +a +10 +110 +4 +1 +1Iclassicalrepartitioned10-102-101.5 +2 +1-52V +0 +XV +5100.50.62 +u +0.65-120.X-11-10Exponential Runge-Kutta Parareal +(a) Error versus Iteration +(b) Parallel Speedup +Figure 4: Parareal configuration with 푁푔 ∈ {1, 2, 3} (all other parameters are in table 3) applied to the nonlinear Schrödinger +equation (16) with initial conditions (21). Subfigure (a) shows how the error at 푦(푡final = 14) evolves as a function of the +Parareal iteration 푘. Subfigure (b) shows how much faster the Parareal algorithms are compared to taking 푁푠 serial steps +with the fine integrator 푓. Increasing 푁푔 improves convergence but also decreases parallel speedup. We again see significant +decrease in speedup due to communication overheads. +converging to the fine solution. We therefore change the accuracy of the coarse propagator  by considering two +additional choices for the number of coarse steps, namely 푁푔 ∈ {1, 2, 3}. By increasing 푁푔 (the number of steps of +푔 in ) we can exchange parallel speedup for a more accurate coarse integrator. In fig. 4 we show error and parallel +speedup as a function of the iteration number 푘 for these three Parareal configurations. As expected, the Parareal +configuration with 푁푔 = 1 does not converge to the fine solution within six iterations. However, as 푁푔 increases we +see that convergence properties improve significantly at the cost of decreased parallel speedup. +In summary, repartitioning is required to prevent instabilities and the accuracy of the coarse integrator must be +increased if we want to resolve high-frequency modes. In the next section we will use linear stability analysis and +linear convergence analysis to more carefully quantify these statements. +5. Linear stability and convergence analysis +In this section we study the linear stability and convergence properties of exponential Parareal and provide a +mathematical foundation for understanding the numerical experiments from section 4. Our analysis is based on the +partitioned Dahlquist equation +푦′ = 휆1푦 + 휆2푦, +푦(0) = 1, +(22) +and follows closely with our previous works [16, 17] that respectively studied implicit-explicit Parareal and reparti- +tioned exponential integrators. We note that for classical Parareal methods, there are many existing works studying +stability and convergence [8, 69, 33, 64] including several that develop rigorous mathematical convergence bounds for +diffusive problems [29, 67, 68]. +Unlike the more general nonlinear system (7), the partitioned Dahlquist equation provides a simple starting point +for mathematical analysis. This arises from the fact that any one-step exponential integrator, including Parareal, reduces +to an iteration of the form +푦푛+1 = 푅(푧1, 푧2)푦푛 +where +푧1 = ℎ휆1, 푧2 = ℎ휆2, +(23) +and ℎ is the method’s stepsize. Consequently, (22) is commonly used to study the stability of both exponential and +implicit-explicit methods [5, 20, 53, 47, 17]; in the case of exponential integrators the term 휆1푦 is exponentiated while +the term 휆2푦 is treated explicitly. It should also be noted that (7) reduces to a system of decoupled, partitioned Dahlquist +equations when the linear and nonlinear operator can be simultaneously diagonalized. Since we are only considering +non-diffusive equations, we assume that 휆1 and 휆2 are purely imaginary. The following table summarizes the relevant +equations. +T Buvoli et al.: Preprint submitted to Elsevier +Page 9 of 33 + +50 +40 +oeedup +30theoreticalachievedS +20 +！ +10 +0 +0 +12 +3 +4 +Iteration k5 +63 +teratio4 +n k511 +610~6Fine Errort +a +-4 +10 +or4 +1 +1I19 +Ng= 310%Ng= 110-80 +12 +1Exponential Runge-Kutta Parareal +Nonlinear system +퐲′ = 퐋퐲 + 푁(퐲) +eig(퐋), eig( 휕퐍 +휕퐲 ) ∈ 푖ℝ +Partitioned Dahlquist +푦′ = 휆1푦 + 휆2푦 +휆1, 휆2 ∈ 푖ℝ. +To estimate stability and convergence properties for a specific nonlinear system, we consider a family of partitioned +Dahlquist equations with continuous 휆1, 휆2 values that respectively enclose the spectrums of the linear operator 퐋 and +the Jacobian of the nonlinear operator 휕푁 +휕퐲 . This rectangular parameter space in the scaled coordinates 푧1, 푧2 is +푍(ℎ) = {푧1 ∈ ℎ[− ̄휆1, ̄휆1], 푧2 ∈ ℎ[− ̄휆2, ̄휆2]} +̄휆1 = 푖휌(퐋), +̄휆2 = 푖 +max +푡∈[푡0,푡final] 휌 +( +휕푁 +휕퐲 (퐲(푡)) +) +(24) +where 휌(⋅) returns the spectral radius and ℎ is the stepsize required by the fine integrator to achieve a desired error +tolerance. Ideally, we would like a Parareal configuration to be stable and rapidly convergent for any (푧1, 푧2) ∈ 푍(ℎ). +Finally, due to the limited stability of exponential integrators on non-diffusive equations, we must consider +repartitioning. If one applies the repartitioning (13) to (15), the equations from the previous table have the following +analogs. +Repartitioned nonlinear system +퐮′ = (퐋 + 휖퐃) +⏟⏞⏟⏞⏟ +̂퐋 +퐮 + (푁(퐮) − 휖퐃퐮) +⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ +̂푁(퐮) +퐋 = 퐔횲퐔−ퟏ, 퐃 = −퐔|횲|퐔−ퟏ +Repartitioned Dahlquist +푦′ = (휆1 − 휖|휆1|) +⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ +̂휆1 +푦 + (휆2 + 휖|휆1|) +⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ +̂휆2 +푦 +Since repartitioning preserves linearity, the iteration (23) for a repartitioned integrator simply becomes +푦푛+1 = 푅(푧1 + 휖|푧1|, 푧2 − 휖|푧1|)푦푛. +(25) +The remainder of this section is organized as follows. In section 5.1 we briefly quantify the parameter ranges that +are pertinent for the discretized nonlinear Schrödinger equation from section 4. Section 5.2 then contains simplified +formulas for the Parareal method on the partitioned Dahlquist equation. In sections 5.3 and 5.4 we use linear analysis +to study the stability and convergence properties of the exponential Parareal iteration. This allows us to quantify the +stability effects of repartitioning, and to understand why high-frequency oscillations cause convergence problems for +Parareal. In section 5.5 we then compare the predictions of linear analysis against the results of our nonlinear numerical +experiments. In section 5.6 we briefly analyze how certain Parareal parameters affect convergence. Lastly, we conclude +with section 5.7 where we discuss the implications of convergence analysis for the solution of partial differential +equations. +5.1. Spectral radius of the nonlinear Schrödinger operators +To analyze the numerical experiments from section 4, we first determine the parameters of the Dahlquist equation +that most closely approximate the discretized nonlinear Schrödinger equation (17). We proceed by bounding the +spectral radius of the linear and nonlinear operators to estimate the rectangular parameter space 푍(ℎ) defined in (24): +• Linear operator. The linear operator 퐋 for the discretized nonlinear Schrödinger equation with an even number +of spatial grid points 푁푥 is +퐋 = diag(−푖퐤2) +for +퐤 = 1 +4[0:푁푥∕2 − 1, −푁푥∕2:−1]푇 . +(26) +Using 푁푥 = 1024 and applying dealiasing we have ̄휆1 = 휌(퐋) = (341∕4)2; dealiasing removes the top one-third +of the highest frequency modes so that only modes −341, … , 341 remain. +• Nonlinear operator Jacobian. Obtaining the exact spectral radius for the nonlinear Jacobian 휕푁 +휕푢 is more involved. +Instead, we estimate its magnitude by assuming there is no coupling between Fourier modes. The continuous +nonlinear operator in physical space is 2푖|푢|2푢, which when applied to a single mode 푢(푥) = 푎푘푒푖푘푥, leads +to 2|푎푘|2푎푘푒푖푘푥. Ignoring mode coupling, the discretized nonlinearity in Fourier space acts like the diagonal +operator diag(2푖|퐮|2). In each of the experiments from section 4, the elements of 퐮 are all bounded above by +1.0001 throughout the temporal domain, so we estimate that ̄휆2 = max푡∈[0,14] 휌 +( +휕푁 +휕퐲 (퐲(푡)) +) +≈ 2. +T Buvoli et al.: Preprint submitted to Elsevier +Page 10 of 33 + +Exponential Runge-Kutta Parareal +Lastly, all the experiments from section 4 use a fine stepsize of ℎ = 14∕216. Therefore, using (24), the nonlinear +Schrödinger equation can be approximately analyzed using the family of Dahlquist equations with scaled parameters +푧1 ∈ 푖[0, 1.6] +and +푧2 ∈ 푖[−4.3, 4.3] × 10−4. +(27) +To avoid imaginary numbers, it is convenient to consider the real-valued dimensions of this parameter region, namely +푟1 ∈ [0, 1.6] +and +푟2 ∈ [−4.3, 4.3] × 10−4. +(28) +We will frequently refer back to these numbers as we study the stability and convergence properties of Parareal on the +nonlinear Schrödinger equation. +5.2. Parareal for the partitioned Dahlquist equation +We now present several formulas that describe the Parareal iteration (4) applied to the Dahlquist equation (22); these +formulas were originally developed in [64] for unpartitioned linear problems. We begin by considering the coarse and +fine propagators (, ) and their underlying integrators (푔, 푓). When applied to (22) these methods reduce to the scalar +iterations +푔: +푦푛+1 = 푅푔(푧1, 푧2)푦푛 +: +푦푛+1 = 푅(푧1, 푧2)푦푛 = 푅푔(훿푧1, 훿푧2)푁푔푦푛 +for +훿 = +푁푓 +푁푔 +푓: +푦푛+1 = 푅푓(푧1, 푧2)푦푛 +: +푦푛+1 = 푅(푧1, 푧2)푦푛 = 푅푓(푧1, 푧2)푁푓 푦푛 +(29) +where ℎ is the stepsize and 푧1 = ℎ휆1 푧2 = ℎ휆2. Note that for 푔 and 푓, 푦푛 corresponds to the 푛th fine step, while for + and , 푦푛 corresponds to the 푛th coarse step (see fig. 1 for an illustration of coarse and fine steps). The Parareal +iteration (4) then reduces to the matrix iteration +퐌퐲푘+1 = (퐌 − 퐌)퐲푘 + 퐛 +(30) +where the vector 퐲푘 = [푦푘 +푗 ] ∈ ℝ푁푝+1 stores the solution at each coarse step, and the matrices 퐌, 퐌 ∈ ℝ푁푝+1,푁푝+1 +and vector 퐛 ∈ ℝ푁푝+1 are +퐌 = +⎡ +⎢ +⎢ +⎢⎣ +퐼 +−푅 +퐼 +⋱ +⋱ +−푅 +퐼 +⎤ +⎥ +⎥ +⎥⎦ +, +퐌 = +⎡ +⎢ +⎢ +⎢⎣ +퐼 +−푅 +퐼 +⋱ +⋱ +−푅 +퐼 +⎤ +⎥ +⎥ +⎥⎦ +, +퐛 = +⎡ +⎢ +⎢ +⎢⎣ +푦0 +0 +⋮ +0 +⎤ +⎥ +⎥ +⎥⎦ +. +(31) +Note that the values 푅 and 푅 are the stability functions from (29) that depend on 푧1 and 푧2. +Next, solving the recurrence relation (30) yields +퐲푘+1 = +푘 +∑ +푗=0 +퐄푗퐌−1 + 퐛, +for +퐄 = 퐈 − 퐌−1 +퐺 퐌−1 +퐹 , +(32) +and we can now interpret the Parareal algorithm as a fixed point iteration that converges to the fine solution +퐲 = 푦0 +[ +1, 푅, 푅2 +, … , 푅 +푁푝 + +]푇 +∈ ℝ푁푝+1. +(33) +Lastly, if we define the error at the 푘th iteration as 퐞푘 = 퐲푘 − 퐲 (i.e. the difference between the Parareal solution and +the serial fine integrator solution), then the error at the 푘th iteration, evolves according to the matrix iteration +퐞푘 = 퐄퐞푘−1. +(34) +To obtain (34), we substitute 퐲푘 = 퐞푘 + 퐲 into (30), left multiply by 푀−1 +퐺 , then simplify using 퐌퐲 = 퐛. In the +following subsections we will use (32) to study stability and (34) to study convergence. +T Buvoli et al.: Preprint submitted to Elsevier +Page 11 of 33 + +Exponential Runge-Kutta Parareal +5.3. Linear stability analysis +Linear stability analysis [72, IV.2] is a well-known technique that is used to determine the types of equations for +which a time integrator is stable (e.g. diffusive or advective). The analysis proceeds by considering the Dahlquist +equation and determining the subset of parameters that lead to a stable iteration. All one-step exponential integrators +applied to the partitioned Dahlquist equation (22) reduce to the iteration (23). The function 푅(푧1, 푧2) is the stability +function of the method and its magnitude must be smaller than or equal to one to guarantee stability. The stability +region of a method contains all the (푧1, 푧2) pairs for which this holds true and is formally defined as +푆 = {(푧1, 푧2) ∈ ℂ2 ∶ |푅(푧1, 푧2)| ≤ 1} . +(35) +For a fixed set of parameters, Parareal is a one-step method that advances the solution by 푁푝 coarse timesteps, or +equivalently, 푁푠 fine timesteps. If 푦푗 denotes the 푗th fine timestep, then Parareal applied to (22) reduces to the iteration +푦(푁푠(푛+1)) = 푅(푁푠푧1, 푁푠푧2)푦(푁푠푛) +where +푧1 = ℎ휆1, 푧2 = ℎ휆2, +(36) +ℎ is the stepsize of the fine integrator 푓, and the stability function 푅 is +푅(푧1, 푧2) = 퐜2 +( 푘 +∑ +푗=0 +퐄푗 +) +퐌−1 +퐺 퐜1, +퐜ퟏ = [1, 0, … , 0]푇 ∈ ℝ푁푝+1, +퐜ퟐ = [0, … , 0, 1] ∈ ℝ푁푝+1. +(37) +This stability function follows directly from (32); 퐜1 is equivalent to 퐛 with 푦0 = 1 and 퐜2 extracts the solution at the +final coarse step. +We now apply linear stability analysis to study Parareal methods with classical and repartitioned ERK integrators. +Since we are interested in non-diffusive problems with 휆1, 휆2 ∈ 푖ℝ, we only consider the two-dimensional stability +region +̂푆 = {(푟1, 푟2) ∈ ℝ2 ∶ |푅(푖푟1, 푖푟2)| ≤ 1} . +(38) +Our aim is to determine if the Parareal method from table 3 is stable for the (푟1, 푟2) region (28) that encloses the +eigenvalues of the discretized nonlinear Schrödinger equation. +In fig. 5 we compare the stability regions of the Parareal configuration from table 3 equipped with either classical +or repartitioned ERK methods. We immediately see that the stability regions associated with classical ERK integrators +only encompass a small subset of the rectangular parameter region (28) and that the rate of instability worsens +significantly as 푘 increases. In contrast, repartitioning greatly expands the stability region of the Parareal method +and the remaining instabilities are sufficiently small that they will not affect the quality of the final solution. Figures 21 +and 22 from appendix D contain additional stability plots that reveal a wider range of 푟2 values. Although repartitioning +greatly improves stability, exponential Parareal is only stable when |푟2| ≪ |푟1|. In other words, the linear term in (7) +must contain the majority of the stiffness. +Overall, linear stability analysis is consistent with the convergence diagrams from figs. 2 and 3 which show that +Parareal with classical ERK methods grows increasingly unstable as the iteration count 푘 increases. From the linear +stability diagrams we also see that the Parareal iteration greatly magnifies the instabilities that are present in the serial +ERK4 integrator. For comparison, the serial ERK4 method is stable across the entire range of (푟1, 푟2) values shown in +fig. 5 and its instability rates near the line 푟2 = 0 are smaller than 1.01; see fig. 3 in [17]. This implies that repartitioning +is important for Parareal even on short timescales where serial exponential integrators do not require it. +5.4. Linear convergence analysis +We now study the convergence rate of the Parareal iteration. We again consider the partitioned Dahlquist equation +(22) and determine the subset of parameters that lead to guaranteed rapid convergence. In section 5.2 we showed that +the difference between the Parareal solution and a serial fine integrator solution evolves according to the iteration (34). +Since Parareal fully converges after exactly 푁푝 iterations, the matrix 퐄 is nilpotent and the convergence rate cannot +be derived from its spectrum. Nevertheless, as originally proposed in [64], monotonic convergence is guaranteed if +‖퐄‖ < 1 since +‖퐞푘+1‖ ≤ ‖퐄‖‖퐞푘‖ < ‖퐞푘‖. +(39) +T Buvoli et al.: Preprint submitted to Elsevier +Page 12 of 33 + +Exponential Runge-Kutta Parareal +Parareal Stability Regions and Instability Factors +푘 = 0 +푘 = 2 +푘 = 4 +푘 = 6 +Classical ERK +Repartitioned ERK +Figure 5: Stability regions for the Parareal configuration from table 3 with classical exponential integrators (top row) and +repartitioned exponential integrators (bottom row). Each column corresponds to a different Parareal iteration 푘. The gray +region is the stability region (38), and color shows the amplification factor |푅(푧1 = 푖푟1, 푧2 = 푖푟2)| outside the stability +region. The 푟1 and 푟2 axis limits on the graphs correspond exactly to (28). +For convergence to occur within a small number of Parareal iterations, we require ‖퐄‖ ≪ 1; for example if ‖퐄‖ < 1∕10 +it will take 10 iterations to reduce the error by 10 digits. In [15] we showed that the ∞-norm of 퐄 is +‖퐄‖∞ = +1 − |푅|푁푝 +1 − |푅| |푅 − 푅|. +(40) +where the values 푅 and 푅 are the stability functions from (29). The ∞-norm is convenient to use since it is both +interpretable and easy to compute. +Using (39) and (40) we define the convergence region ∞ to be the set of all (푧1, 푧2) pairs for which the ∞-norm +of 퐄 is smaller than one +∞ = {(푧1, 푧2) ∈ ℂ ∶ ‖퐄(푧1, 푧2)‖∞ < 1} . +(41) +Since we are only considering non-diffusive equations with 휆1, 휆2 +∈ 푖ℝ, we will study the two-dimensional +convergence region +̂∞ = {(푟1, 푟2) ∈ ℝ ∶ ‖퐄(푖푟1, 푖푟2)‖∞ < 1} . +(42) +Note that the matrix 퐄 does not depend on the iteration 푘 so a single convergence region pertains to a Parareal +configuration with an arbitrary 퐾. +We now apply linear convergence analysis to understand why high-frequency oscillations cause problems for +Parareal and why increasing the number of coarse steps 푁푔 improves convergence. We again consider the three Parareal +configurations from fig. 4 with 푁푔 ∈ {1, 2, 3} and all other parameters from table 3. We are primarily interested to +see if the convergence regions of these three Parareal configurations enclose the rectangular region (28). In fig. 6, we +present the Parareal convergence regions; the red rectangles in each plot show the largest rectangular subset of (28) +that can be enclosed inside each convergence region. +The first observation is that the convergence regions near (푟1 = 0, 푟2 = 0) grow approximately linearly in size +with respect to 푁푔. This observation follows directly from (40) since increasing 푁푔 makes the coarse propagator more +accurate, therefore decreasing the quantity |푅−푅| (See remark 1 in appendix E). Overall, linear convergence analysis +T Buvoli et al.: Preprint submitted to Elsevier +Page 13 of 33 + +2 +0 +-2104 +103 +102X10 +4 +4 +2107 +106 +105-4 +0 +0.8 +1.6 +1101 +100×10-4 +4 +2 +2-2 +-4 +00.8 +r11.62 +0 +-2×10 +.4 +4 +2-4 +00.8 +r11.62 +0 +-2×10 +4 +200.8 +r11.62 +-2X10 +.4 +4 +2-4 +00.8 +r11.6-24 +2-4 +00.8 +r11.6-2×10 +4 +2-4 +00.8 +r11.6-2×10 +4 +2-4 h +00.8 +r11.6-2×10 +4 +2-4 +00.8 +r11.6Exponential Runge-Kutta Parareal +퐍퐠 = ퟏ +퐍퐠 = ퟐ +퐍퐠 = ퟑ +Magnified 푟2 axis +Magnified 푟2 axis +Magnified 푟2 axis +Figure 6: Convergence regions (42) for the three Parareal configurations from fig. 4. The columns represent different +choices for 푁푔 and the bottom row shows a magnified 푟2 axis compared with the top row. Color represents the ∞-norm +of the matrix 퐄, from (32), and the red rectangles are the largest rectangular subset of the 푟1 and 푟2 range from (28) +that can be contained inside the convergence region. The 푟1-coordinate of the labeled point corresponds to the width of +the rectangular subset rounded to three digits; it is defined as 푟max +1 += max휌 subject to {푟1 = 휌, 푧2 ∈ [−2ℎ, 2ℎ]} ⊆ ̂퐶∞ for +ℎ = 14∕216. +confirms that increasing 푁푔 leads to a Parareal configuration that will resolve a larger number of high-frequency +temporal components. +The second observation is that the convergence regions are small and fail to fully enclose (28). More precisely, +while all of the 푟2 range is inside the convergence region, less than twenty percent of the 푟1 range is included, even +when 푁푔 = 3. However, recall that in fig. 4 we saw that the Parareal configuration with 푁푔 = 3 was able to accurately +converge to the fine solution; we will explore this fact in more detail in section 5.5. +Our third and final observation involves convergence rates for small, fixed (푟1, 푟2) and follows directly from +remark 1. Specifically, if we let 푞 be the order of the coarse integrator, then, for fixed (푟1, 푟2), ‖퐄‖∞ = (1∕푁푔 +푞). +Therefore increasing 푁푔 will also increase the Parareal convergence rate. +5.5. Validating convergence results for the nonlinear Schrödinger equation +We now validate how closely the predictions of linear convergence analysis, made using the Dahlquist parameters +(27), align with the results from fig. 4. The nonlinear Schrödinger equation was spatially discretized using a Fourier +spectral discretization that represents the solution as the sum of 푀 Fourier modes, such that +푢(푥, 푡) = +푀∕2−1 +∑ +푛=−푀∕2, +퐚푛(푡)푒푖푘푥∕4. +(43) +T Buvoli et al.: Preprint submitted to Elsevier +Page 14 of 33 + +0.2 +r10.0 +30.2200.40.60.0508-0.05 +00.1.1 +r10.20.3200.073, 0.05-0.05 +0.1 +r10.20.320.133, 0）0.05-0.05 +0.1 +r10.20.320 (0188,00.05-0.05 +01 +r10.20.320.073, 0510-45 +00.1 +r10.20.320.133, 0510-45 +00.1 +r10.20.3200188, 0)510-45 +00.Exponential Runge-Kutta Parareal +푁푔 +푟max +1 +Convergent 퐚푛(푡) +1 +.073 +푛 ∈ [−73, 73] +2 +.133 +푛 ∈ [−99, 99] +3 +.188 +푛 ∈ [−118, 118] +Figure 7: Monotonically convergent Fourier coefficients, as predicted by linear convergence analysis, for the three Parareal +configurations from fig. 4 with 푁푔 ∈ {1, 2, 3}. The set of convergent mode indices is defined as {푛 ∈ ℤ ∶ 푟1(푛) ≤ 푟max +1 +} for +푟1(푛) = ℎ푛2∕16 and ℎ = 14∕216. The table contains 푟푚푎푥 +1 +values (the 푥 coordinates of the labeled points in fig. 6) along with +the indices of the corresponding convergent coefficients. The blue line in the plot shows 푟1(푛), the solid gray horizontal +lines correspond to the three 푟max +1 +values, and the pairs of vertical dashed lines are the upper and lower bounds for the +predicted convergent coefficients. +The Fourier coefficients 퐚푛(푡) evolve according to (7) and (17) with 퐲 = [퐚0, … , 퐚푀∕2−1, 퐚−푀∕2, … , 퐚−1]푇 . Therefore, +the differential equation that governs the 푛th coefficient is +̇퐚푛 = 휆1(푛)퐚푛 + [푁(퐲)] +훼(푛)−푀∕2 +where +휆1(푛) = 푖푛2∕16, +(44) +[푁(퐲)]푗 is the 푗th component of the nonlinearity, and 훼(푛) = [1 + 푛 + 푀∕2] (mod 푁). To conduct linear analysis, we +replace the coupled nonlinearity with the decoupled linear term 휆2퐚푛, with 휆2 ∈ 푖[−2, 2]. It then follows that a Parareal +iteration with fine timestep ℎ will monotonically converge to the solution of 퐚푛(푡) only if (ℎ휆1(푛), ℎ휆2) is inside the +convergence region ̂퐶∞. In other words, linear convergence analysis predicts that Parareal will only monotonically +converge to the subset of Fourier coefficients 퐚푛(푡) for which 푛 satisfies the set inequality +{푟1 = 푟1(푛), 푟2 ∈ [−2ℎ, 2ℎ]} ⊆ ̂∞ +for +푟1(푛) = ℎ푛2∕16. +(45) +Ignoring stability considerations, all other 퐚푛(푡) will remain at the accuracy achieved by the coarse integrator until +푘 → 푁푝. We can simplify the condition (45) by introducing +푟max +1 += max +휌 +subject to +{푟1 = 휌, 푟2 ∈ [−2ℎ, 2ℎ]} ⊆ ̂∞, +(46) +which is the width of the largest rectangle that includes the entire 푟2 range and is enclosed by the convergence region. +The values of 푟max +1 +for the three Parareal configurations considered in fig. 4 are the 푟1-coordinates of the labeled points +in fig. 6. Using (46) it follows immediately that the condition (45) is equivalent to the inequality +푟1(푛) < 푟max +1 +. +(47) +In fig. 7 we present a table of the 푟max +1 +values for the three parareal configurations from fig. 4, along with the resulting +estimates of the monotonically convergent Fourier modes. Then in fig. 8 we validate these estimates by comparing the +Fourier coefficients 퐚푛(푡) obtained using the Parareal iteration to those obtained using the serial ERK4 integrator. +Overall we see that linear convergence analysis very accurately predicts the convergent Fourier coefficients. Moreover, +we see that the accuracy of all Fourier coefficients with an 푟1(푛) that was outside of the convergence region did not +improve substantially beyond what was achieved using the coarse integrator. +5.6. Convergence regions for additional Parareal configurations +Convergence regions depend on all the Parareal parameters from table 2 and on the repartitioning constant 휌 +from (15). Here we investigate the effects of changing the coarse integrator  and the repartitioning constant 휌. +Figure 9 presents convergence regions for Parareal configurations with  ∈ {ERK1, ERK2, ERK3, ERK4} and all +other parameters taken from table 3. We see that using a higher-order coarse integrator results in a larger convergence +region. This follows directly from (40) since the increased accuracy of a high-order coarse propagator decreases the +T Buvoli et al.: Preprint submitted to Elsevier +Page 15 of 33 + +0 +ficient inoI +73 +lex n99 1180.073- +- +--0.133- +-- +--- +- +- +■0.188-H- +L- +L +-中 +- +- +- +-- +-ri(n- +-+ +- +-I +-118-99I +-73 +COeExponential Runge-Kutta Parareal +퐍퐠 = ퟏ +퐍퐠 = ퟐ +퐍퐠 = ퟑ +Error in Fourier Coefficient 퐚푛(14) +Error in Fourier Coefficient 퐚푛(14) +Error in Fourier Coefficient 퐚푛(14) +Error Norm vs Iteration +Error Norm vs Iteration +Error Norm vs Iteration +Coarse (퐾 = 0) +Parareal (1 ≤ 퐾 ≤ 5) +Parareal (퐾 = 6) +Fine Propagator (퐾 = 푁푝) +Figure 8: Plots describing the same numerical experiment as the one from fig. 4. Each column corresponds to a Parareal +configuration with a different value of 푁푔 and the colors represent different Parareal iteration numbers. Top Row: Error +in the Fourier coefficients 퐚푛(푡) of the solution (43) at 푡 = 14 for Parareal with 퐾 ∈ {0, … , 6}. The black line corresponds +to the fine propagator  run in serial. The two vertical dotted lines in each plot are the upper and lower bounds for the +convergent spatial modes as predicted by linear analysis and are identical to those shown in fig. 7. Bottom Row: Error norm +between the reference solution and the Parareal method at 푡 = 14. These plots are identical to the one shown in fig. 4. +quantity | − |. Therefore, replacing a low-order coarse propagator with a higher-order one, provides another way to +improve convergence for high-frequency temporal modes. Naturally, any convergence gains must be weighed against +the decrease in parallel speedup (5) caused by the more expensive high-order coarse propagator. +Next, we briefly discuss how the repartitioning parameter 휌 from (15) affects convergence; recall that 휌 is the angle +(in radians) that the eigenvalues of the linear operator are rotated into the left-half plane. Figure 10 contains convergence +regions for the Parareal configuration from table 3 with repartitioning parameters 휌 ∈ {0, 휋∕256, 휋∕64, 휋∕16}. When +no repartitioning is applied (i.e. 휌 = 0) an exponential integrator will exactly solve a Dahlquist equation (22) with +휆2 = 0. Therefore, the Parareal convergence region for 휌 = 0 extends infinitely along the line 푟2 = Im(ℎ휆2) = 0. Any +amount of repartitioning destroys the exactness of the integrator along this line. In practice this is not important since +imposing 휆2 = 0 is equivalent to forcing the nonlinearity 푁(푡, 푦) in (2) to be zero. It should be noted however, that +increasing the repartitioning parameter leads to a more subtle contraction of the overall convergence region. Therefore, +to maximize convergence for high-frequency information, one should select the smallest repartitioning constant that +ensures stability. +5.7. Implications of convergence analysis for solving non-diffusive PDEs +As demonstrated by linear analysis, Parareal only converges rapidly to equation components that are not overly +oscillatory in time. There are many factors affecting the number of high-frequency temporal modes present in a spatially +discretized partial differential equation. High-order dispersive derivatives (e.g. 푢푥푥푥 or 푖푢푥푥) possess continuous +T Buvoli et al.: Preprint submitted to Elsevier +Page 16 of 33 + +0 +icient- +: +73 +index n1701010° +JO. +n- +- +- +- +- +- +--10-1- +170-73 +coeff0 +icientindex 99 +n1701010° +JO. +n-- +- +- +- +- +-- +- +-10- +-1170-99 +coeff0 +icientindex 118 +n1701010° +JO. +n- +-- +- +- +- +- +- +- +-- +-h- +- +- +-10-1170 -11.8 +coeffK士10Err +10Fine SollutionError10 +r +01-一K士10Err +10Fine Sollution Err10 +r +01-一K士10Err +10Fine Solljtion kror10 +r +01-一Exponential Runge-Kutta Parareal + = ERK1 + = ERK2 + = ERK3 + = ERK4 +Figure 9: Convergence regions for the Parareal configuration from table 3 with different coarse integrators . +휌 = 0 +휌 = 휋∕256 +휌 = 휋∕64 +휌 = 휋∕16 +Figure 10: Convergence regions of the Parareal configuration from table 3 with four repartitioning constants 휌 from (15). +spectrums with large imaginary eigenvalues. The extent to which the continuous spectrum causes convergence +problems will depend on the choice of spatial discretization. Non-diffusive discretizations, such as Fourier pseudo- +spectral methods, will present the greatest challenge since the discretized linear operators will also have purely +imaginary eigenvalues. Using fine spatial grids will also increase the total number of high-frequency components. +Finally, there is the question of whether high-frequency information is necessary for obtaining accurate solutions. This +will depend on the initial condition, the length of the integration window, and the characteristics of the problem. PDEs +or initial conditions that cause rapid spectral broadening within the integration window will be the most challenging +to solve using Parareal. In summary, limited convergence for high-frequency temporal components will manifest as +inaccuracies in high-frequency spatial modes of the solution. +For the generic initial value problem (7), we can extend the linear analysis developed in this section to estimate the +convergence properties of a given Parareal configuration. We start by making the following two assumptions: +1. The linear operator 퐋 is diagonalizable and (휆푛, 퐯푛) are the 푛th eigenvalue and eigenvector. This allows us to +express the solution as a linear combination of the eigenvectors, 퐲(푡) = ∑푀 +푛=0 푎푛(푡)퐯푗. +2. The linear operator contains the majority of the stiffness such that 휌(퐋) ≫ 휌( 휕푁 +휕푦 ). +We then bound the spectrum of the linear and nonlinear operators and define the region +(ℎ) = {푟1 ∈ ℎ[0, 푐1], 푟2 ∈ ℎ[−푐2, 푐2]} +푐1 = 휌(퐋), +푐2 = max +푡 +휌 +( +휕푁 +휕퐲 (퐲(푡)) +) +. +(48) +To proceed we must first ensure that a given Parareal configuration is sufficiently stable for all (푟1, 푟2) in (ℎ). If this +holds true, we then determine the largest rectangular subset of (ℎ) that: (i) is enclosed by the Parareal convergence +region (42) and (ii) contains the entire 푟2 range. The width of this region is +푟max +1 +(ℎ) = max +휔 +subject to +{푟1 = 휔, 푟2 ∈ ℎ[−푐2, 푐2]} ⊆ ̂∞. +(49) +The value of 푟max +1 +(ℎ) will depend on all Parareal parameters except for 퐾. Finally, we estimate that a Parareal iteration +with fine stepsize ℎ will monotonically converge to the fine integrator solution of any coefficient 푎푛(푡) where 푛 satisfies +|ℎ휆푛| < 푟max +1 +(ℎ). +(50) +T Buvoli et al.: Preprint submitted to Elsevier +Page 17 of 33 + +.04 +r10.00 +80.2200.40.60.03 ~8-0.03 +00.04 +r10.00 +80.2200.40.60.03 ~8-0.03 +00.0 +r140.08200.03 -0.03 +00.0 +r140.08200.03 -0.03 +00.0 +r140.08200.03 -0.03 +00.0 +r140.08200.03~-0.03 +00.0 +r140.08200.03 -0.03 +00.0 +r140.08200.03 -0.03 +00.0 +r140.08200.03 -0.03 +00.0 +r140.08200.03 -0.03 +0Exponential Runge-Kutta Parareal +All the remaining coefficients will retain the accuracy achieved with the coarse integrator and fail to converge for small +iteration count 푘. Although these estimates are rooted in linear theory, our results in section 5.5 demonstrate that this +approach has the potential to accurately predict convergence for nonlinear problems. +6. Higher-dimensional numerical experiments +We now demonstrate that it is possible to achieve meaningful parallel speedup using an exponential Parareal method +on higher-dimensional non-diffusive equations. We conduct two additional numerical experiments in which we solve +the dispersive Kadomtsev-Petviashvili (KP) equation and the hyperbolic Vlasov-Poisson (VP) equation. Both PDEs are +equipped with periodic boundary conditions and discretized in space using a Fourier spectral method. Since analytical +solutions are not known, we compute a reference solution using ERK4 with a very small timestep. The error is then +defined as +‖퐲ref − 퐲method‖∞∕‖퐲ref‖∞. +(51) +where 퐲 represents the solution in physical space. Below we describe the equations, their initial conditions, and the +corresponding numerical parameters. +The Kadomtsev-Petviashvili (KP) equation is +(푢푡 + 6푢푢푥 + 푢푥푥푥 +) +푥 + 3휎2푢푦푦 = 0 +(52) +where 휎2 = −1 leads to KPI that models thin films with large surface tension, while 휎2 = 1 leads to KPII that models +water waves with small surface tension [11]. Both equations admit the soliton solution 푢(푥, 푦, 푡) = 2푝2sech(푝(푥−4푝2푡)) +where 푝 is a free variable. The stability of a perturbed soliton depends on the sign of 휎2, with the KPI solution being +unstable and the KPII solution being stable [28, 45, 46]. +For any well-localized solution in 푥, the KP equation can be expressed in evolution form as +푢푡 + 6푢푢푥 + 푢푥푥푥 + 3휎2휕−1푢푦푦 = 0 +where +휕−1푓 = 1 +2 +[ +∫ +푥 +−∞ +푓(푠)푑푠 − ∫ +∞ +푥 +푓(푠)푑푠 +] +. +(53) +To ensure smoothness in time, the initial condition must satisfy the following equality at 푡 = 푡0 +∫ +∞ +−∞ +푢푦푦(푥, 푦, 푡)푑푥 = 0. +(54) +Any initial condition that does not satisfy this constraint will produce a solution that satisfies (54) for 푡 > 푡0, leading +to a temporal discontinuity at 푡0 [1]. +For our numerical experiment we consider the KPI equation equipped with periodic boundary conditions on the +domain 푥 ∈ [−8휋, 8휋], 푦 ∈ [0, 8휋]. We spatially discretize using 972 grid points in 푥 and 750 grid points in 푦, and +dealias using the standard 3/2 rule. We integrate the equation in Fourier space where the operator 휕−1 is equivalent +to the Fourier multiplier −푖∕푘푥; Note that when 푘푥 = 0 this mode is singular. However, for any initial condition that +satisfies (54) we can simply set this multiplier to zero; for more general initial conditions, numerical regularization +must be added [49]. +As in [45] we select our initial condition to be a soliton with a perturbed phase +푢(푥, 푦, 푡 = 0) = 2sech2 ((푥 + 4휋) + 훿 cos(푦∕4)) , +훿 = 1∕5, +(55) +and integrate the equation to time 푡final = 4. Our initial condition satisfies (54), therefore, no regularization is needed. +As shown in fig. 11, the perturbation is unstable and leads to the formation of a two-dimensional soliton. +The hyperbolic Vlasov-Poisson (VP) equation is +푓푡 + 푣푓푥 + 퐸(푥, 푡)푓푣 = 0, +for +퐸푥(푥, 푡) = −1 + ∫ +∞ +−∞ +푓(푥, 푣, 푡)푑푣, +(56) +and describes the evolution of charged particles in an electric field [35]. Our numerical experiment is based on the +bump-on-tail experiment from [21]. Specifically, we equip the VP equation with periodic boundary conditions on the +T Buvoli et al.: Preprint submitted to Elsevier +Page 18 of 33 + +Exponential Runge-Kutta Parareal +Figure 11: Solution of the KPI equation (53) at 푡 = 4 with initial conditions (55). +domain 푥 ∈ [0, 20휋], 푣 ∈ [−8, 8], and spatially discretize using a 1024 point Fourier discretization in both 푥 and 푣. +Our initial condition is +푓(푥, 푣, 푡 = 0) = +( +0.9 +√ +2휋 +푒−푣2∕2 + 0.2 +√ +2휋 +푒−2(푣−4.5)2 +) ( +1 + +4 +100 cos(0.3푥) +) +(57) +and the solution is integrated to time 푡final = 50. To preserve a diagonal linear operator we solve the equation in +physical 푣 space and Fourier 푥 space; see (67) in appendix F. As shown in fig. 12 the bump-on-tail initial condition +excites modes that lead to complex dynamics. +Figure 12: Solution of the Vlasov-Poisson equation (56) at 푡 = 50 for the initial condition (57). +6.1. Parareal parameter selection and experiment overview +The Parareal configurations we selected to solve the KP and VP equations are described in table 4. For both +equations, we considered multiple configurations that differ only in the number of coarse steps 푁푔. We vary this +parameter to demonstrate the improved convergence properties associated with larger 푁푔 values. For the fine integrator +푓 we always selected ERK4 and set the total number of steps 푁푠 so that a fully-converged Parareal method produces a +T Buvoli et al.: Preprint submitted to Elsevier +Page 19 of 33 + +0 +-10 +c1020 +1020H04 +n10 +5y +0-20660. +2 +0 +2 +-8/ +7 +-6 +-4240 +505 +0.330200 +1002 +7 +4 +aExponential Runge-Kutta Parareal +(a) Kadomtsev-Petvaishvili (KP) +Parareal Parameters +푁푝 +8192 +푁푓 +32 +푓 +ERK3 +퐾 +1, … , 28 +푁푠 +218 +푁푔 +{1, 2, 3} +푔 +ERK4 +푟max +1 +values +푁푔 = 1 +푁푔 = 2 +푁푔 = 3 +푟max +1 +(ℎ) +0.0642 +0.112 +0.155 +(b) Vlasov-Poisson (VP) +Parareal Parameters +푁푝 +2048 +푁푓 +64 +푓 +ERK4 +퐾 +1, … , 12 +푁푠 +217 +푁푔 +{2, 3} +푔 +ERK3 +푟max +1 +values +푁푔 = 2 +푁푔 = 3 +푟max +1 +(ℎ) +0.074 +0.102 +Table 4 +Parareal parameters used to solve the KP equation (53) with initial conditions (55) and the Vlasov-Poisson equation (56) +with initial conditions (57). In the right-most tables we present the associated 푟max +1 +values. All 푟max +1 +values are computed +using the stepsize ℎ = 푡final∕푁푠, the repartitioning parameter 휌 = 휋∕128, and by assuming that the stiffness in the nonlinear +term is negligible such that 푐2 = 1 in (48). +highly accurate solution. The remaining parameters were then determined by balancing the achievable parallel speedup +with the size of the convergence regions. +The results for the numerical experiments in this section will be summarized in three plots: (i) error versus iteration +퐾, (ii) parallel speedup versus iteration 퐾, and (iii) error versus run-time. In the error versus run-time plots we +will compare the efficiency of the Parareal iteration to that of the coarse and fine ERK integrators run in serial. All +experiments were performed using 32-core Haswell nodes on the Cray XC40 Cori at the National Energy Research +Scientific Computing Center. For the VP equation we collected timing results by running the Parareal iteration on 64 +nodes (2048 compute cores). The full KP experiment requires 256 nodes (8096 compute cores) which exceeded our +available computational resources. Therefore, we ran the Parareal iteration in serial on a single node to determine the +convergence curves, and then extrapolated the achievable speedup from a smaller experiment where we solved the KP +equation on the shortened interval 푡 ∈ [0, 1] using 64 nodes. +Lastly, for each Parareal configuration we compute the convergent spatial modes as predicted by the linear stability +analysis from section 5.7. For simplicity, we assume that any stiffness in the nonlinear term is negligible so that +convergence depends exclusively on the eigenvalues of the linear operator (i.e. 푐2 = 0 in (48)). +6.2. Kadomtsev-Petvaishvili – results and discussion +The KP equation is challenging to solve because the third-order derivative term leads to a linear operator with large +imaginary eigenvalues. As we have seen in sections 5.5 and 5.7, these eigenvalues determine the degree of temporal +oscillation present in the spatial Fourier coefficients of the solution. Therefore, we must select a Parareal configuration +whose convergence region contains at least a large subset of these eigenvalues. Our proposed configurations are +described in table 4a. The choice of total steps 푁푠 ensures that a fully-converged Parareal iteration will produce a +solution with an accuracy of 1.2 × 10−9. Moreover, we selected ERK3 as the coarse integrator because it is stable at +sufficiently large stepsizes (see fig. 23) and the convergence region for an ERK3, ERK4 pairing is larger than that of +an ERK2, ERK4 pairing (see fig. 9). +We can estimate the convergent spatial modes for each choice of 푁푔 using the stability analysis developed in +section 5.7. The eigenvalues of the linear operator 퐋 are +퐋(푘푥, 푘푦) = +{ −푖(휔푥푘푥)3 +푘푥 = 0 +−푖(휔푥푘푥)3 + 푖 +(휔푦푘푦)2 +휔푘푥 +푘푥 ≠ 0 +(58) +where 푘푥 and 푘푦 are integer Fourier wavenumbers and 휔푥 = 2휋∕퐿푥 = 1∕8, 휔푦 = 2휋∕퐿푦 = 1∕4. The convergent +spatial modes lie inside the region + = {(푘푥, 푘푦) ∈ ℤ2 ∶ ℎ|퐋(푘푥, 푘푦)| < 푟max +1 +(ℎ)} +for +ℎ = 2−16, +(59) +T Buvoli et al.: Preprint submitted to Elsevier +Page 20 of 33 + +Exponential Runge-Kutta Parareal +where the values of 푟max +1 +depend on 푁푔 and are contained in table 4a. In fig. 13 we overlay the region  onto the +linear operator 퐋 and the Fourier transformed final solution. This allows us to estimate which spatial modes will +be accurately computed by each Parareal configuration. Although none of the regions  enclose the entire (푘푥, 푘푦) +domain, the coefficients for the highest-frequency spatial modes are small and only need to be resolved if we require +an extremely accurate solution. +(a) Magnitude of the scaled KP linear operator eigenvalues +(b) KP Solution at 푡 = 8 (Fourier transformed in 푥,푦) +Figure 13: Magnitude of the linear operator eigenvalues (fig. 13a) and final solution in Fourier space (fig. 13b) for the +KP equation. The axis in both plots are the Fourier wavenumbers in 푥 and 푦. The black contours shows the boundaries +of the region  from (60) that encloses all convergent spatial modes as predicted by linear convergence analysis. The +dotted, dashed, and solid line respectively correspond to the region  for Parareal configurations with 푁푔 = 1, 푁푔 = 2, +and 푁푔 = 3. From linear analysis, we expect that each Parareal iteration will converge monotonically for all spatial modes +inside its corresponding region . +In fig. 14 we show convergence, speedup, and efficiency results for the exponential Parareal methods applied to the +KP equation. We divide our discussion of the results into three parts: +1. Convergence. No exponential Parareal method converged monotonically across the entire 푘 range. Instead they +exhibited a rapid reduction in error during the first few iterations before entering a plateau of slow convergence. +This behavior is expected, since none of the Parareal convergence regions enclose the full spectrum of the +discretized KP linear operator. As predicted by linear analysis, increasing 푁푔 improves convergence, and the +Parareal configuration with 푁푔 = 3 is able to obtain a solution that is comparable in accuracy to the serial +fine integrator. In contrast, the Parareal configurations with 푁푔 ∈ {1, 2} do not resolve a sufficient number +of high-frequency spatial modes to achieve fine error within 27 iterations; nevertheless, both methods improve +the coarse solution by multiple orders of magnitude. These results are analogous to those for the NLS equation +shown in fig. 4. +2. Parallel speedup. In contrast to the one-dimensional NLS equation, the KP equation is more computationally +expensive to integrate over a single timestep. This is due to the fact that a right-hand-side evaluation now requires +multiple two-dimensional discrete Fourier transforms. This reduces the significance of communication costs, and +we now see very good agreement between the theoretical and achieved parallel speedup. In summary, though +the penalties incurred due to communication costs are mildly visible (notice that all dashed lines are slightly +below the solid lines in fig. 14b), the theoretical speedup estimate (5) provides a realistic measure for real-world +performance of the Parareal iteration. +3. Efficiency. Perhaps the most important result is the error versus time plot, from which we see that all three Parareal +configurations are able to compute high-accuracy solutions significantly faster than the serial ERK methods. +T Buvoli et al.: Preprint submitted to Elsevier +Page 21 of 33 + +00.51-0.500.51Ng=1 ---- Ng=2 +T— Ng = 3-1 +-11 +-0.5100 +2 +100 +300-200K +I +0.2 +2 +h +T010100 +0F 90 +6 +K +0.4200-300-200.00 +0100 +200 +300-200)nl) +0g10( +0.10100 +04) +5 - +-0200-300-200.00 +0Exponential Runge-Kutta Parareal +(a) Error versus Iteration +(b) Parallel Speedup versus Iteration +(c) Error versus Time +Figure 14: Numerical results for the KP equation using the three Parareal configurations from table 4, that are differentiated +with different colored lines. (a) Solution error at the final time 푡 = 4 as a function of Parareal iteration. (b) Theoretical +speedup (5) and achieved speedup from the numerical experiment. (c) Efficiency diagram comparing the Parareal +configurations to the serial coarse and fine integrators. +Moreover, despite their failure to converge to the fine solution within 28 iterations, the Parareal configurations +with 푁푔 ∈ {1, 2} remain the fastest methods for obtaining moderately less accurate solutions. We summarize +this in the table below: +푁푔 +Error Tol at 푘 = 28 +Speedup compared to serial ERK4 +1 +2.7 × 10−7 +9.09x +2 +1.3 × 10−8 +10.15x +3 +1.8 × 10−9 +11.71x +Lastly we note that fig. 14b shows both the efficiency for the Parareal method using the theoretical speedup +(dashed lines) from (5) and the achieved speedup (solid lines). Unlike our one-dimensional experiments, we +now see good agreement between these two qualities and the corresponding lines look nearly identical. +6.3. Vlasov-Poisson – results and discussion +The Vlasov-Poisson equation does not contain high-order spatial derivatives, therefore it is possible to select +Parareal parameters that simultaneously offer good Parallel speedup and convergence properties. Our proposed Parareal +configurations are described in table 4b. The choice of total steps 푁푠 ensures that a fully-converged Parareal iteration +will produce a solution with an accuracy of 6.6 × 10−8 (see fig. 23). +We again apply linear analysis to estimate the convergent spatial modes for each choice of 푁푔. The eigenvalues of +the linear operator 퐋 are +퐋(푘푥, 푣) = 푖휔푥푘푥푣 +for +푘푥 ∈ ℤ, 푣 ∈ , +and + = {−8 + 16푗∕푁푣}푁푣−1 +푗=0 +(60) +where 푣 represents a discrete grid point on the domain [−8, 8], 푘푥 is the Fourier wavenumber in 푥, and 휔푥 = 2휋∕퐿푥 = +1∕10. The convergent spatial modes lie inside the region + = {(푘푥, 푣) ∈ ℤ ×  ∶ ℎ|퐋(푘푥, 푣)| < 푟max +1 +(ℎ)} +for +ℎ = 50∕217, +(61) +where the values of 푟max +1 +are contained in table 4b. Because we are only Fourier transforming in the 푥 direction, it is +important that our Parareal configuration accurately computes all the components in the 푣 domain since we cannot +assume spectral decay in the physical 푣 direction. +In fig. 15 we overlay the convergence region  onto the linear operator 퐋 and the transformed final solution. The +convergence regions for Parareal configurations with both 푁푔 = 1 and 푁푔 = 2 enclose the entire discrete (푘푥, 푣). We +T Buvoli et al.: Preprint submitted to Elsevier +Page 22 of 33 + +teratioFine Errora +rror +10 +64 +-3 +1I +t19 +Ng= 3100.N10F +2eratiol1 k2%10Speed +30 +S +2040 +up50... theoreticalachievedF +2上 +6 +10 +I11) +ive Tir10 +ne +(sec21 +a +rror +-6 +10'4 +1I +t100ERK3 +ERK410 +10°1( +RelatExponential Runge-Kutta Parareal +note that the largest diagonal element of the scaled linear operator ℎ퐋 is 0.0693, therefore the convergence region for +the Parareal configuration with 푁푔 = 1 just barely encloses the eigenvalues since 푟max +1 +(ℎ) = 0.0740. +In fig. 16 we show convergence, speedup, and efficiency plots for the exponential Parareal method applied to the +VP equation. We again divide our discussion of the results into three parts: +1. Convergence. The Parareal method with 푁푔 = 2 failed to converge, while the method with 푁푔 = 3 displayed +monotonic convergence and achieved the fine error tolerance after eight iterations. The failure of convergence +for 푁푔 = 2 is likely due to several reasons. First, linear analysis is not guaranteed to provide an accurate +prediction for all nonlinear equations. Moreover, linear analysis predicts that the Parareal with 푁푔 = 2 is only +just barely convergent, so a larger safety margin may be required to properly predict convergence on nonlinear +problems. Lastly, it is also possible that the divergent iteration is due to instabilities. Specifically, our assumption +that the nonlinear term is completely non-stiff may be inaccurate due to the presence of the term −푓푣 in the +nonlinearity. Fortunately, modestly increasing 푁푔 resolves these issues and leads to a stable, monotonically +convergent Parareal iteration. +2. Parallel Speedup. As with the KP equation, we see very good agreement between the theoretical and achieved +parallel speedup. We can again see very minor penalties due to communication (notice that all dashed lines are +slightly below the solid lines), however the differences are even smaller than those for the KP equation. This +follows from the fact that the cost per timestep is more expensive for the VP equation than for the KP equation. +3. Efficiency. The Parareal configuration with 푁푔 = 3 was able to obtain a solution 24 times faster than the serial +ERK4 method. +푁푔 +Error Tol at 푘 = 8 +Speedup compared to serial ERK4 +3 +6.6 × 10−8 +24x +This substantial gain in speedup compared to the KP experiment is made possible by the lack of high-oscillatory +temporal components. This allowed us to run the coarse integrator at a significantly larger stepsize relative +to the fine integrator. For comparison, the coarse integrator of the Parareal configurations with 푁푔 = 3 for +the KP and VP equations, were respectively run with a stepsize that was 10.6 and 21.3 times larger than the +fine integrator. Overall this experiment demonstrates the potential for very significant speedup when applying +exponential Parareal to accurately solve hyperbolic equations. +7. Conclusions and future work +In this paper we applied exponential integrators within the Parareal iteration and presented linear analysis that +can be used to study stability and convergence properties of the resulting methods on non-diffusive equations. We +then demonstrated the ability of exponential Parareal methods to achieve significant parallel speedup on multiple non- +diffusive partial differential equations. +We draw two main conclusions from this work. First we showed that repartitioning is essential for obtaining a +Parareal configuration that is stable on stiff non-diffusive equations. Second, through linear analysis we were able to +better understand the convergence characteristics of the Parareal iteration in the absence of diffusion. Specifically we +saw that the Parareal iteration achieves fine integrator accuracy for low-frequency (i.e. non-stiff) oscillatory modes +and coarse integrator accuracy for high-frequency (i.e. stiff) oscillatory modes. When solving non-diffusive partial +differential equations this phenomenon makes it impossible to guarantee rapid convergence for high-frequency spatial +modes. Therefore, exponential Parareal is best suited for non-diffusive equations and initial conditions that do not cause +rapid spectral broadening. +To the best of the authors’ knowledge, this is the first paper to investigate the usage of exponential integrators +within a parallel-in-time framework. Our initial results look promising as we have demonstrated the ability to achieve +parallel speedup using exponential Parareal on both hyperbolic and dispersive equations. Nevertheless, there are still +many avenues that require further exploration. In particular all of the numerical experiments presented in this paper +involve diagonal linear operators that greatly simplify the computation of the exponential 휑-functions. In future work +we plan to study exponential Parareal integrators in the more general setting with non-diagonal linear operators and +examine the resulting effects on computational performance and parallel speedup. +T Buvoli et al.: Preprint submitted to Elsevier +Page 23 of 33 + +Exponential Runge-Kutta Parareal +(a) Magnitude of the VP linear operator eigenvalues +(b) VP Solution at 푡 = 50 (Fourier transformed in 푥 only) +Figure 15: Magnitude of the linear operator eigenvalues (fig. 15a) and the transformed final solution (fig. 15b) for the +Vlasov-Poisson equation. The axes in both plots represent the Fourier wavenumber in 푥 and the spatial variable 푣. The +region  from (61) for both the Parareal configurations from table 4 encloses the entire discrete (푘푥, 푣) domain; therefore +no black contours are shown on the plots. +(a) Error versus Iteration +(b) Parallel Speedup +(c) Error versus Time +Figure 16: Numerical results for the VP equation using the two Parareal configurations from table 4, that are differentiated +with different colored lines. (a) Solution error at the final time 푡 = 50 as a function of Parareal iteration. (b) Theoretical +speedup (5) and achieved speedup from the numerical experiment. (c) Efficiency diagram comparing the Parareal +configurations to the serial coarse and fine integrators. +Acknowledgements +The work of Buvoli was funded by the National Science Foundation, Computational Mathematics Program DMS- +2012875. 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Method coefficients +This appendix contains the Tableaus described in section 3.1 for the exponential Runge-Kutta integrators that are +used in this paper. We use the abbreviation 휑푖,푗 = 휑푖(푐푗ℎ퐋) for the 휑-functions. +• ERK1: first-order exponential Euler method (10) +0 +휑1 +• ERK2: second-order method from [20] +0 +1 +휑1 +휑1 − 휑2 +휑2 +• ERK3: third-order method from [20] +0 +1 +2 +1 +2휑1,2 +1 +−휑1 +2휑1 +휑1 − 3휑2 + 4휑3 +4휑2 − 8휑3 +−휑2 + 4휑3 +• ERK4: fourth-order method from [53] +0 +1 +2 +1 +2휑1,2 +1 +2 +1 +2휑1,2 − 휑2,2 +휑2,2 +1 +휑1 − 2휑2 +0 +2휑2 +휑1 − 3휑2 + 4휑3 +2휑2 − 4휑3 +2휑2 − 4휑3 +−휑2 + 4휑3 +T Buvoli et al.: Preprint submitted to Elsevier +Page 27 of 33 + +Exponential Runge-Kutta Parareal +Error versus Stepsize – Initial Condition (19) +Error versus Computational Time – Initial Condition (19) +Figure 17: Convergence diagram (left) and precision diagram (right) for the exponential Runge-Kutta methods listed +in appendix A run on the nonlinear Schrödinger equation (16) with initial condition (19). Colored lines correspond to +repartitioned integrators (rERK) and gray lines correspond to unmodified exponential integrators (ERK) – repartitioning +has no effect on this problem. The black crosses on the ERK3 and ERK4 method respectively correspond to the stepsizes +of the coarse and fine integrators for the Parareal method described in table 3. +Error versus Stepsize – Initial Condition (20) +Error versus Computational Time – Initial Condition (20) +Figure 18: Identical to fig. 17 except we are now considering the initial condition (20). +B. Nonlinear Schrödinger Serial ERK Results +We solve the nonlinear Schrödinger equation (16) using the serial ERK methods from from appendix A using 2푝 +timesteps where 푝 = 7, 8, … , 19. In figs. 17 to 19 we show accuracy and convergence diagrams for the initial conditions +(19) to (21), respectively. +C. NLS solution for initial condition (21) +Figure 20 shows the two different NLS solutions arising from the initial conditions (19) and (21). +T Buvoli et al.: Preprint submitted to Elsevier +Page 28 of 33 + +Stepsizeh10-110 +10rERK +rERK +rERK + rEBK4 +3 +2 +1a +10 +ErrorERK4 +ERK3 +ERK2 +ERK14 +.2 +1 +10 +1I +t +七11 +- +- +- +- +TT1010210100 +Time +（se10110210 +& +10a +10 +Error +-6= +- +-4 +.2 +1 +10 +1I +t +七11 +- +- +- +- +TT10°11 +- +- +-102TT10°Stepsizeh10-110 +10rERK +rERK +rERK + rEBK4 +3 +2 +1a +10 +ErrorERK4 +ERK3 +ERK2 +ERK14 +.2 +1 +10 +1I +t +七11 +- +- +- +- +TT1010210100 +Time +（se10110210 +& +10a +10 +Error +-6= +- +-4 +.2 +1 +10 +1I +t +七11 +- +- +- +- +TT10°11 +- +- +-102TT +-10Exponential Runge-Kutta Parareal +Error vs Stepsize – Initial Condition (21) +Ng = 1 +Ng = 2 +Ng = 3 +Error vs Computational Time – Initial Condition (21) +Ng = 1 Ng = 2 Ng = 3 +Figure 19: Identical to fig. 17 except we are now considering the initial condition (21). +NLS Solution – Initial Condition (19) +NLS Solution – Initial Condition (21) +Figure 20: Solutions of the nonlinear Schrödinger equation (16) for the initial conditions (19) and (21). +D. Additional Stability Plots +Figures 21 and 22 contain additional stability plots for the Parareal configuration table 3 that show different (푟1, 푟2) +ranges than fig. 5. +E. Remark regarding convergence region scaling +Remark 1. Stability regions grow approximately linearly in 푁푔 for small (|푟1|, |푟2|). To show this, we first assume +that we are in a regime where (|푟1|, |푟2|) is small so that the coarse and the fine integrator both exhibit asymptotic +error properties. If we construct the coarse propagator  using 푁푔 steps of a 푞th order integrator, then +|푅| = |||푒푖(푟1+푟2)||| +  +( +|푟1|+|푟2| +푁푔 +)푞 +< 1 + 퐶1 +( +|푟1|+|푟2| +푁푔 +)푞 +, +(62) +|푅 − 푅| =  +( +|푟1|+|푟2| +푁푔 +)푞 +< 퐶2 +( +|푟1|+|푟2| +푁푔 +)푞 +. +(63) +T Buvoli et al.: Preprint submitted to Elsevier +Page 29 of 33 + +T +714010510X-10010rERK +rERK +rERK + rEBK4 +3 +2 +1a +r +Error +10ERK4 +ERK3 +ERK2 +ERK14 +1 +10 +t +七10%Stepsize102h10-11010°a +10 +Error +-61 +- +-4 +.2 +1 +10 +1I +t +七11 +- +- +- +- +TT10°100 +Time +（se11 +- +- +-102101TT +-10210 +& +10714010510X-100Exponential Runge-Kutta Parareal +Parareal Stability Regions and Instability Factors +퐾 = 0 +퐾 = 2 +퐾 = 4 +퐾 = 6 +Classical ERK +Repartitioned ERK +Figure 21: Additional stability regions for the Parareal configuration from table 3 with classical exponential integrators +(top column) and repartitioned exponential integrators (bottom column). The gray region is the stability region (35), and +color shows the amplification factor |푅(푧1 = 푖푟1, 푧2 = 푖푟2)| outside the stability region where the method is unstable. These +figures show a wider range of 푟2 values than fig. 5. +The norm of the error matrix can be bounded above by +‖퐄‖∞ = +1 − |푅|푁푝 +1 − |푅| |푅 − 푅| < 푁푝퐶2 +( +|푟1|+|푟2| +푁푔 +)푞 ++  +( +|푟1|+|푟2| +푁푔 +)푞 +. +(64) +Convergence is guaranteed if ‖퐄‖∞ < 1, which is equivalent to +|푟1| + |푟2| < +1 +푁푔퐶2푁푝 ++  +( +|푟1|+|푟2| +푁푔 +)푞 +. +(65) +Ignoring the higher order terms, the size of the region |푟1| + |푅2| < (퐶2푁푔푁푝)−1 grows linearly in 푁푔. +F. Spatially discretized Vlasov-Poisson equation +For notational simplicity we represent the discrete VP solution as the matrix 퐟 where 퐟푗푘 approximates the +continuous solution at the grid point (푣푗, 푥푘). Next we define the scaled Fourier wavenumber vectors +퐤푣 = 휋 +퐿푣 +[0, … , 푁푣∕2 − 1, −푁푣∕2, … , −1], +퐤푥 = 2휋 +퐿푥 +[0, … , 푁푥∕2 − 1, −푁푥∕2, … , −1], +(66) +for 퐿푣 = 16, 퐿푥 = 20휋, and the ℝ푁푣,푁푥 matrices 퐊푣 +푗푘 = 퐤푣 +푗, 퐊푥 +푗푘 = 퐤푥 +푘. If 푣(⋅) and 푥(⋅) represent the discrete Fourier +transform in 푣 and 푥, then the transformed variable ̂퐟 = 푥(퐟) satisfies +푑 +푑푡 +̂퐟푗푘 = 푣푗푖퐤푥 +푘̂퐟푗푘 + 푥 +( +퐄.* 휕퐟 +휕푣 +) +푗푘 +휕퐟 +휕푣 = −1 +푥 +( +−1 +푣 +( +푖퐊푣.*푣 +( +̂퐟 +))) +(67) +T Buvoli et al.: Preprint submitted to Elsevier +Page 30 of 33 + +2 +0 +-0.041015 +10100.08 +0.041025 +1020-0.08 +00.8 +1.6 +r1105 +1002 +0 +-0.040.08 +0.04-0.08 +00.8 +r11.62 +0 +-0.040.08 +0.04-0.08 +00.8 +r11.62 +-0.040.08 +0.04-0.08 +00.8 +r11.62 +0 +-0.040.08 +0.04-0.08 +00.8 +r11.62 +0 +-0.040.08 +0.04-0.08 +00.8 +r11.62 +-0.040.08 +0.04-0.08 +00.8 +r11.62 +0 +-0.040.08 +0.04-0.08 +00.8 +r11.62 +-0.040.08 +0.04-0.08 +00.8 +r11.6Exponential Runge-Kutta Parareal +Parareal Stability Regions and Instability Factors +퐾 = 0 +퐾 = 2 +퐾 = 4 +퐾 = 6 +Classical ERK +Repartitioned ERK +Figure 22: Additional stability regions for the Parareal configuration from table 3 with classical exponential integrators +(top column) and repartitioned exponential integrators (bottom column). The grey region is the stability region (35), and +color shows the amplification factor |푅(푧1 = 푖푟1, 푧2 = 푖푟2)| outside the stability region where the method is unstable. These +figures show a narrrower range of 푟1 and a wider range of 푟2 than fig. 5. +where .* denotes the Hadamard product, and the discrete electric field 퐄푗푘 = 퐞푘 ≈ 퐸(푥푘, 푡) is +퐞 = −1 +푥 +( +퐤−퐱.* +( +퐛 + Δ푣 +푁푣 +∑ +푗=1 +̂퐟푗푘 +) +⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ +푥(−1+∫ 20휋 +0 +푓(푥,푣,푡)푑푣) +) +, +for +퐛 = 푥(−[1, … , 1]푇 ) ∈ ℝ푁푥, +퐤−푥 +푘 += +{ +0 +푗 = 1, +1∕퐤푥 +푘 +푗 > 1 ∈ ℝ푁푥, +Δ푣 = 16∕푁푣. +(68) +Note that the integral term ∫ 20휋 +0 +푓(푥, 푣, 푡)푑푣 is treated using the trapezoidal rule, which convergences exponentially +on periodic domains. +G. ERK convergence diagrams for KP and Vlassov-Poisson +In fig. 23 we show convergence diagrams for the serial ERK methods applied to the KP and VP equations. The plots +also contain black crosses that indicate the step-sizes of the coarse and fine integrators for the Parareal configurations +described in section 6. +T Buvoli et al.: Preprint submitted to Elsevier +Page 31 of 33 + +2 +0 +-0.041015 +10100.08 +0.041025 +1020-0.08 +00.2 +0.4 +r1105 +1002 +0 +-0.040.08 +0.04-0.08 +00.2 +r10.42 +0 +-0.040.08 +0.04-0.08 +00.2 +r10.40 +-0.040.08 +0.04-0.08 +00.2 +r10.40 +-0.040.08 +0.04-0.08 +00.2 +r10.42 +0 +-0.040.08 +0.04-0.08 +00.2 +r10.42 +0 +-0.040.08 +0.04-0.08 +00.2 +r10.40 +-0.040.08 +0.04-0.08 +00.2 +r10.40 +-0.040.08 +0.04-0.08 +00.2 +r10.4Exponential Runge-Kutta Parareal +Error versus Stepsize – KP equation (53) +Ng = 1 +Ng = 2 +Ng = 3 +Fine Integrator +Error versus Stepsize – VP Equation (56) +Coarse (Ng = 2) +Coarse (Ng = 3) +Fine Integrator +Figure 23: Serial ERK convergence diagrams for the KP and VP equations. We can related these plots to the Parareal +configurations described in table 4. Specifically, the labeled black crosses at large timesteps correspond to the stepsizes of +the coarse integrator, while the black cross on the ERK4 method at the smallest stepsize corresponds to the stepsize of +the fine integrator. +T Buvoli et al.: Preprint submitted to Elsevier +Page 32 of 33 + +10-10rERK +rERK +rERK + rEBK4 +3 +2 +1Relativ +10 +10ERK4 +ERK3 +ERK2 +ERK1TTT +TTT +-2 +Error +10- +-Stepsize102h1010-810rERK +rERK +rERK + rEBK4 +3 +2 +1Relativ +10ERK4 +ERK3 +ERK2 +ERK1-3Crror +e +10- +-10Stepsize10°102 +hTTT10- \ No newline at end of file diff --git a/89E2T4oBgHgl3EQfQAYH/content/tmp_files/load_file.txt b/89E2T4oBgHgl3EQfQAYH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4530586d7d4a79becd556c94ea413a820ce95cc5 --- /dev/null +++ b/89E2T4oBgHgl3EQfQAYH/content/tmp_files/load_file.txt @@ -0,0 +1,1479 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf,len=1478 +page_content='Exponential Runge-Kutta Parareal for Non-Diffusive Equations ⋆ Tommaso Buvolia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Michael Minionb aTulane University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' New Orleans,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' LA 70118,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' USA bLawrence Berkeley National Lab,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' CA 94720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' USA A R T I C L E I N F O Keywords: Parareal,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Parallel-in-time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Exponen- tial Integrators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Linear Stability Anal- ysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Convergence Analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Non- Diffusive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Hyperbolic 2010 MSC: 65L04,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 65L05,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 65L06,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 65L07 A B S T R A C T Parareal is a well-known parallel-in-time algorithm that combines a coarse and fine propagator within a parallel iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' It allows for large-scale parallelism that leads to significantly reduced computational time compared to serial time-stepping methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' However, like many parallel-in- time methods it can fail to achieve parallel speedup when applied to non-diffusive equations such as hyperbolic systems or dispersive nonlinear wave equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This paper explores the use of exponential integrators within the Parareal iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Exponential integrators are particularly interesting candidates for Parareal because of their ability to resolve fast-moving waves, even at the large stepsizes used by coarse propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This work begins with an introduction to expo- nential Parareal integrators followed by several motivating numerical experiments involving the nonlinear Schrödinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' These experiments are then analyzed using linear analysis that approximates the stability and convergence properties of the exponential Parareal iteration on nonlinear problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The paper concludes with two additional numerical experiments involving the dispersive Kadomtsev-Petviashvili equation and the hyperbolic Vlasov-Poisson equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' These experiments demonstrate that exponential Parareal methods can achieve significant parallel speedup on different types of non-diffusive equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Introduction Time integrators [37, 72, 12] are numerical methods that solve an initial value problem by sequentially advancing the solution via a series of discrete timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' For more than half a century, these methods have proven invaluable for modeling a range of dynamical processes appearing in both science and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In a typical calculation one iteratively applies a time integrator to evolve a system over thousands or even millions of timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Therefore, the total computational cost is not just that of a single timestep, but rather the combined cost of applying the method sequentially over the full temporal domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The sequential nature of classical time-stepping methods has come under increasing scrutiny in light of modern parallel hardware like multicore processors, massively parallel high performance computing systems, and specialized accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' For more than two decades, these advancements have spurred the development of new parallel-in-time (PinT) methods [44, 54, 24, 27, 30] that distribute the full temporal domain over a large number of computational nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Perhaps the most well-known PinT method is the Parareal algorithm [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Parareal consists of a parallel iteration that combines a fine propagator (a computationally expensive and accurate integrator) with a coarse propagator (a computationally cheap and less accurate integrator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The aim of Parareal is to obtain the solution of the fine propagator at a similar computational cost to that of the coarse propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Parareal has proven effective for accelerating the solution of diffusive equations [26, 71, 58, 52] and its theoretical convergence properties are well understood in the presence of diffusion [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In contrast, non-diffusive equations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' hyperbolic systems or dispersive nonlinear wave equations) introduce significant numerical difficulties that lead to slow convergence or instabilities in the Parareal iteration [8, 69, 31, 64, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Though numerous modifications have been proposed [19, 22, 23, 25, 32, 51], the resulting methods introduce additional complexities that make them less applicable to all types of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In this work we combine exponential integrators [43] with Parareal and demonstrate, both theoretically and experimentally, that this pairing can provide parallel speedup on non-diffusive equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We focus specifically on ⋆This work was funded by the National Science Foundation, Computational Mathematics Program DMS-2012875 ∗Corresponding author tbuvoli@tulane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='edu (T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Buvoli);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' mlminion@lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='gov (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Minion) ORCID(s): T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 1 of 33 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03764v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='NA] 10 Jan 2023 aExponential Runge-Kutta Parareal the non-diffusive, semilinear initial value problem 퐲′ = 퐋퐲 + 푁(푡, 퐲), 퐲(푡0) = 퐲0 (1) where the eigenvalues of both 퐋 and 휕푁 휕퐲 (the Jacobian of 푁(푡, 퐲)) are purely imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We were motivated to consider exponential integrators because they treat the linear component 퐋 exactly, granting them the ability to accurately resolve fast moving waves even at coarse stepsizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This property is beneficial because the convergence of Parareal on non- diffusive problems depends critically on accuracy differences between the coarse and fine solver [64, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' However, special care must be taken when applying exponential integrators on non-diffusive equations since the methods are classically unstable [17, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In fact, we will demonstrate that the repartitioning strategy introduced in [17] is essential for obtaining stable exponential Parareal methods on stiff non-diffusive problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The organization of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In section 2 and section 3 we respectively introduce Parareal and exponential integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In section 4 we motivate this paper by presenting several numerical experiments involving the one-dimensional nonlinear Schrödinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Then, in section 5 we introduce analytical tools for understanding the convergence and stability properties of a Parareal configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Lastly, in section 6 we present additional numerical experiments that demonstrate exponential Parareal’s ability to achieve parallel speedup on higher- dimensional hyperbolic and dispersive wave equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Parareal Introduction In this section we describe the Parareal algorithm [54], present a formula for parallel speedup, and provide a complete table of all Parareal parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We begin by supposing that one seeks an accurate, numerical solution to the initial value problem 퐲′(푡) = 푓(퐲(푡)), 퐲(푡0) = 퐲0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (2) If computational cost can be neglected, then an accurate numerical integrator \ue232, such as a high-order method with small timesteps, should be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' However, this will not always be practical since the time to run the calculation can become prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Therefore, we often settle for a less accurate integrator \ue233, such as a low-order method that is run using larger timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Can the situation be improved with access to parallel computational hardware?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The Parareal method is a parallel iteration that combines a coarse propagator \ue233 with a fine propagator \ue232, and converges to the solution of \ue232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Provided that the iteration can be efficiently parallelized and that the convergence rate is sufficiently high, then the computational time needed to run Parareal is similar to that of running the coarse propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In the following subsections, we explore the algorithm in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Method definition Let \ue232 and \ue233 be two one-step methods that are respectively called the coarse and fine propagators;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' it is assumed that \ue232 is more computationally expensive to apply than \ue233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Next, suppose that we want to approximate (2) at a discrete set of time points using the fine propagator, such that 푦푛+1 = \ue232(푦푛), 푛 = 0, … , 푁푝, (3) where 푦푛 ≈ 푦(푡푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The Parareal algorithm converges to (3) by taking a provisional solution, 푦0 푛 ≈ 푦(푡푛), that is usually computed by a serial application of the coarse propagator \ue233, and then correcting it via the iteration 푦푘+1 푛+1 = \ue233(푦푘+1 푛 ) + \ue232(푦푘 푛) − \ue233(푦푘 푛), { 푛 = 0, … , 푁푝, 푘 = 0, … , 퐾 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (4) In order to run the Parareal iteration, it is necessary to store and iteratively update the solution values along the entire time interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The key property of the Parareal iteration is that the fine integrator \ue232 can be applied in parallel on 푁푝 processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' To further clarify this point, we show pseudocode for the Parareal iteration (4) in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 2 of 33 Exponential Runge-Kutta Parareal Parareal Pseudocode 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' % provisional solution 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' for n = 0 : 푁푝 − 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 푦0 푛+1 = \ue233(푦0 푛) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' % Parareal iteration 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' for k = 0 : K - 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' parfor j = 0 : 푁푝 − 1 % parallel loop 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 퐹푗 = \ue232(푦푘 푗 ) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' for j = 0 : 푁푝 − 1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 푦푘+1 푗+1 = \ue233(푦푘+1 푗 ) + 퐹푗 − \ue233(푦푘 푗 ) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' return 푦퐾 푁푝 Table 1 Pseudocode for the Parareal iteration (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The fine integrator (colored in red) can run in a parallel loop, while the loops containing the coarse propagator (colored in blue) are serial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Pseudocode for more efficient pipelined implementations are contained in [7, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Parallel speedup Parallel speedup is defined as the ratio between the computational time for running a serial algorithm and its parallel equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' For Parareal we compute speedup by dividing the computational cost of the sequential fine integrator (3) by the computational cost of the Parareal iteration (4) when run using 푁푝 processors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Let the cost for a single step of the fine propagator \ue232 and the coarse propagator \ue233 be 퐶\ue232 and 퐶\ue233, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The cost of performing 퐾 Parareal iterations is the sum of the cost of the predictor, 푁푝퐶\ue233, plus the additional cost of each iteration which, neglecting communication, is 퐾(퐶\ue232 + 퐶\ue233);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' see [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In summary, the serial cost for computing (3) is 퐶푠 = 푁푝퐶\ue232 and a parallel cost for (4) is 퐶푝 = 푁푝퐶\ue233 + 퐾(퐶\ue232 + 퐶\ue233).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' If we let 훼 = 퐶\ue233∕퐶\ue232, then the parallel speedup is 푆 = 퐶푠 퐶푝 = 푁푝 푁푝훼 + 퐾(1 + 훼).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (5) Lastly we remark the speedup formula will change if one considers more elaborate parallelization strategies such as those presented in [7, 9, 4, 63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Selecting the coarse and fine propagators A user can select any pair of one-step methods to be the coarse and fine propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' A common approach, which we will use in this work, is to set the coarse propagator \ue233 equal to 푁푔 steps of an inexpensive one-step method 푔 and the fine propagator \ue232 equal to 푁푓 steps of an expensive integrator 푓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Both \ue232 and \ue233 must advance the solution by the same amount;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' therefore, if we let ℎ be the stepsize of 푓, then the stepsize of 푔 must be ℎ푁푓∕푁푔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' If we use the notation 푀휅(휂) to denote 휅 steps of a the method 푀 run with stepsize 휂, then the fine and coarse propagators are \ue232 = 푓 푁푓 (ℎ) and \ue233 = 푔푁푔(ℎ푁푓∕푁푔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (6) In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 1 we illustrate the the resulting coarse and fine grids for the coarse and fine propagators (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Since the fine propagator \ue232 is now 푁푓 steps of the method 푓, a Parareal method that converges to the solution of \ue232 applied over 푁푝 steps is also converging to the solution of 푓 applied over total of 푁푠 = 푁푓푁푝 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Throughout this work we will frequently characterize Parareal in terms of (푓, 푁푓), (푔, 푁푔), and 푁푠 instead of \ue232, \ue233, and 푁푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' A complete table of parameters As we have seen, the Parareal algorithm has a large number of free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In table 2 we make a complete list of parameters that are relevant to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' As we note in the table, the integer variables 푁푝, 푁푠, 푁푏 and 푁푓 are related by the equation 푁푠 = 푁푏푁푝푁푓 and therefore the user can only select three of these variables with the fourth being automatically determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 3 of 33 Exponential Runge-Kutta Parareal G f(h) f(h) f(h) g(3h) F Figure 1: An illustration of the coarse and fine propagators (blue arrow for coarse and red arrows for fine), the coarse temporal grid (large, black squares), and fine temporal grid (small, white circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This illustration depicts the following parameters: the time interval has been divided into 12 total timesteps (푁푠 = 12), the coarse propagator \ue233 that takes a single step of 푔 (푁푔 = 1), the fine propagator \ue232 that takes three steps of 푓 (푁푓 = 3), and the resulting Parareal iteration requires 4 processors (푁푝 = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Interdependent User Defined Parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='(A user must select two in such a way that all three variables are integers) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Variable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Meaning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='푁푝 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Number of processors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='푁푠 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Number of fine steps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='푁푓 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Number of RK steps in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Interdependency ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='푁푠 = 푁푝푁푓 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Independent User Defined Parameters (A user must select all) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Variable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Meaning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='푓 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='RK method used in \ue232 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='푔 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='RK method used in \ue233 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='푁푔 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Number of RK steps in \ue233 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='퐾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Number of Parareal iterations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Dependent Parameters ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Variable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Meaning ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Definition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='ℎ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Time step for serial method ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='푡final∕푁푠 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='\ue233 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Coarse propagator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='푁푔 steps of RK method 푔 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='\ue232 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Fine propagator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='푁푓 steps of RK method 푓 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='푁푇 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Total number of fine steps per block ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='푁푠∕푁푏 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='푐푔 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Cost of method per step in \ue233 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='User defined ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='푐푓 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Cost of method per step in \ue232 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='User defined ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='퐶\ue232 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Cost of \ue232 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='푁푓푐푓 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='퐶\ue233 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Cost of \ue233 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='푁푔푐푔 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Table 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Parareal parameters names and definitions that are relevant to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Exponential integrators The aim of this work is to study Parareal methods where the coarse and fine propagators are exponential Runge- Kutta methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In this section we provide an introduction to exponential integrators and discuss their stability properties on non-diffusive equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Exponential integrators [43] are a class of numerical methods for solving the semilinear initial value problem 퐲′ = 퐋퐲 + 푁(푡, 퐲), 퐲(푡0) = 퐲0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (7) In the past two decades, they have proven highly efficient for solving stiff systems and can offer certain advantages over both fully-implicit and linearly-implicit methods [36, 48, 55, 57, 40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The main idea behind exponential integrators T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 4 of 33 Exponential Runge-Kutta Parareal is to consider the exact solution to (7), namely 퐲(푡0 + ℎ) = 푒ℎ퐋퐲0 + ∫ 푡0+ℎ 푡0 푒(푡0+ℎ−휏)퐋푁(휏, 퐲(휏))푑휏, (8) and replace the nonlinear term 푁(휏, 퐲(휏)) with an explicit polynomial approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Different approximations lead to different families of exponential integrators, including exponential linear multistep methods [10], Runge-Kutta methods [20, 53, 42, 56, 13], and general linear methods [61, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Since 푁(휏, 푦(휏)) is approximated with a polynomial, the coefficients of all exponential integrators depend on a subset of the exponential functions 휑0(ℎ퐋) = 푒ℎ퐋 and 휑푗(ℎ퐋) = ∫ 1 0 푒(1−푠)ℎ퐋푠푗푑푠 (푗 ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (9) At each timestep an exponential integrator requires matrix-vector products with the 휑-functions of the linear operator 퐋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' For many problems this can be done efficiently using a number of different algorithms [6, 18, 39], including those based on squaring methods [50, 3, 2], contour integration [48, 70], Krylov-subspaces [40, 41, 59, 60, 34], and parallel rational approximations [38, 66, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Exponential Runge-Kutta methods Exponential Runge-Kutta (ERK) methods are one-step methods that approximate the solution to (2) by taking a linear combination of stage values at each timestep .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The simplest ERK integrator is the exponential Euler method that is obtained by replacing 푁(휏, 퐲(휏)) in (8) with the constant approximation 푁(푡푛, 퐲푛), yielding 퐲푛+1 = 휑0(ℎ퐋)퐲푛 + 휑1(ℎ퐋)ℎ푁(푡푛, 퐲푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (10) More generally, an 푠-stage ERK method is 푌푖 = 휑0(ℎ푐푗퐋)퐲푛 + 푖−1 ∑ 푗=1 푎푖푗(ℎ퐋)푁(푐푗, 푌푗), 푖 = 1, … , 푠, (11) 퐲푛+1 = 휑0(ℎ퐋)퐲푛 + 푠 ∑ 푗=1 푏푗(ℎ퐋)푁(푐푗, 푌푗) (12) where 푎푖푗(ℎ퐋) and 푏푗(ℎ퐋) are functions that include linear combinations or products of the 휑-functions (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' By applying the identity 휑0(ℎ퐋)퐲푛 = 퐲푛+휑1(ℎ퐋)퐋퐲푛 one can rewrite the equations (11) and (12) in terms of 휑푗(ℎ퐾) for 푗 ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' this can be advantageous both for method analysis and implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Lastly, like classical RK methods, ERK methods can be represented using the Butcher Tableau 푐1 0 푐2 푎21 0 ⋮ ⋮ ⋱ ⋱ 푐푠 푎푠,1 … 푎푠,푠−1 0 푏1 … 푏푠−1 푏푠 where the coefficients 푎푖푗 and 푏푗 are now matrix functions of the linear operator ℎ퐋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In this work, we will consider ERK methods of orders one to four from [20, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We name these methods ERK1, ERK2, ERK3, and EKR4, and list their Tableaus in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Stability and repartitioning for non-diffusive equations Since exponential integrators treat the linear operator 퐋 exactly, we would expect that they offer significantly improved stability properties compared to explicit integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' While this is true for diffusive operators, the situation is more nuanced when 퐋 has purely imaginary eigenvalues [17, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In particular, both exponential and explicit integrators have similarly sized stability regions, but the magnitude of the instabilities is often very small for exponential T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 5 of 33 Exponential Runge-Kutta Parareal integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Therefore, unlike explicit methods, exponential integrators can still produce usable solutions on stiff non- diffusive equations so long as the total number of timesteps is not overly large [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In [17] we proposed a strategy that stabilizes exponential integrators by repartitioning the right-hand-side of (7) using perturbed linear and nonlinear operators ̂퐋 and ̂ 푁.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This enables long-time simulations with exponential integrators and also removes instabilities when the underlying equation focuses energy into unstable modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The perturbed operators are formed by respectively adding and subtracting a diffusive operator 퐃 such that ̂퐋 = 퐋 + 휖퐃 and ̂ 푁(푡, 퐲) = 푁(푡, 퐲) − 휖퐃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (13) In short, we add damping to the linear operator ̂퐋 and excitation to the nonlinear operator ̂ 푁(푡, 퐲).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The differential equation (7) can then be written in terms of the perturbed operators as 퐲′ = ̂퐋퐲 + ̂ 푁(푡, 퐲).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (14) Therefore, an exponential integrator that solves the repartitioned equation (14) is simultaneously solving (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' However, instead of treating 퐋 exactly and approximating 푁(푡, 퐲), a repartitioned integrator treats ̂퐋 exactly and approximates ̂ 푁(푡, 퐲).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The advantage of repartitioning is that the exponential integrator now possesses a large stability region for a continuous range of small 휖 values [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' If 퐋 is diagonalizable, such that 퐋 = 퐔횲퐔−1, and we select 퐃 = −퐔|횲|퐔−1 and 휖 = 1 tan(휋∕2 − 휌) for 휌 ∈ (0, 휋∕2), (15) then we rotate all the eigenvalues of a non-diffusive linear operator 퐋 by 휌 degrees into the left-half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In other words, the eigenvalues of ̂퐋 all lie on the wedge 푟푒푖휃 for 푟 ≥ 0 and 휃 ∈ {휋∕2 + 휌, 3휋∕2 − 휌}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Many other choices for 퐃 are possible (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' low-order, even spatial derivatives), however in [17] we proposed (15) because of its convenience when analyzing the stability effects of repartitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In summary, exponential integrators exhibit mild instabilities for stiff non-diffusive equations that can be eliminated through re-partitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In sections 4 and 5 we will show that these instabilities are greatly exacerbated by the Parareal iteration, and that repartitioning is essential for obtaining stable, exponential Parareal methods for solving stiff non- diffusive equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Motivating numerical experiments In this section we present three numerical experiments that highlight key properties of exponential Parareal integrators applied to non-diffusive equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Later, in section 5 we will see how linear stability analysis and linear convergence analysis can be used to more rigorously quantify our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' All three numerical experiments involve the one-dimensional nonlinear Schrödinger (NLS) equation 푖푢푡 + 푢푥푥 + 2|푢|2푢 = 0 (16) on the domain 푥 ∈ [−4휋, 4휋] with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We discretize the equation in space using a 1024 point Fourier spectral method that is dealiased using the classical 3∕2 rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The equation is then integrated in Fourier space where the derivative operators are diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This results in the semilinear equation (7) with 퐋 = diag(−푖퐤2) and 푁(푡, 퐲) = 2푖\ue232(\ue232−1(퐲) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' * abs(\ue232−1(퐲))), (17) where 퐤 is a vector of Fourier wavenumbers and \ue232 denotes the discrete Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' To ensure the classical stability of exponential integrators we also consider the repartitioning (13) and (14) where the diffusive operator 퐃 and 휖 are selected according to (15) such that 퐃 = diag(−퐤2), 휖 = 1 tan(휋∕2 − 휌) and 휌 = 휋 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (18) In addition to investigating stability and convergence, we also compare the theoretical speedup of the Parareal iteration to its real-world performance on a distributed memory system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' To do this, we implemented the exponential T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 6 of 33 Exponential Runge-Kutta Parareal 푁푝 2048 푁푓 32 푓 ERK4 퐾 1, … , 6 푁푠 216 푁푔 1 푔 ERK3 Table 3 Parareal parameters used for the nonlinear Schrodinger equation (16) with initial conditions (19) and (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Parareal method as part of the open source package LibPFASST1 and performed the numerical experiment on the Cray XC40 Cori at the National Energy Research Scientific Computing Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Our first experiment uses the initial condition 푢(푥, 푡 = 0) = 1 + 1 100 cos(푥∕4), (19) integrated out to time 푡 = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Repartitioning is not required for serial ERK methods on this short time-scale, and the convergence curves for classical and repartitioned exponential integrators look identical (see convergence plots in appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We now consider two Parareal methods: one with classical ERK integrators, and the other with repartitioned ERK integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' To obtain a high-accuracy solution, we select ERK4 as the fine integrator 푓, and 푁푠 = 216 as the total number of fine steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' from the serial ERK convergence diagrams in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 17 we see that a fully converged Parareal method will yield the solution with an error of 3 × 10−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Next we must select a coarse integrator 푔 that is stable at large stepsizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Though it may seem tempting to select ERK1 because it is the least expensive method per timestep, its poor stability leads us to choose the more stable ERK3 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Lastly, we select 푁푔 = 1 and 푁푓 = 32;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' this implies that the coarse propagator \ue233 consists of a single step of 푔, while the fine propagator \ue232 consists of 32 steps of 푓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' With these parameters the serial coarse propagator has an accuracy of 2 × 10−1, which is approximately eight orders of magnitude less than the serial fine integrator (see the black crosses in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' A complete list of the Parareal parameters used for this experiment is contained in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The computation was distributed on sixty-four, 32-core Intel Haswell nodes that provided a total of 2048 compute cores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 2 we show how the error of the solution obtained by the Parareal iteration evolves as a function of the iteration number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We also show the corresponding theoretical and achieved parallel speedup, along with a space-time plot of the NLS solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The results demonstrate that repartitioning is essential for obtaining a convergent Parareal iteration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' moreover, by using repartitioned ERK methods, Parareal is able to obtain a high-accuracy solution up to 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='5 times faster than a serial ERK4 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In contrast unmodified exponential integrators lead to a divergent Parareal iteration whose error increases monotonically for iteration number 푘 greater than two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Our timing results also reveal a practical challenge that can occur when applying a PinT method on a distributed memory system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Using the Parareal configuration from table 3, we were only able to achieve a speedup factor of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' approximately one quarter less than the theoretical speedup factor of 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='5 predicted by (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This difference is due to unaccounted communication overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In fact, (5) will only be accurate if the time required to compute a single step of the propagator \ue233 is significantly greater than the time for transferring the solution vector between two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Although this does not hold true for the one-dimensional NLS equation, the exponential Parareal method is neverthless able to provide a high-accuracy solution an order of magnitude faster than the serial ERK4 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Moreover, in section 6 we will see that theoretical speedup very accurately predicts achievable speedup on more computationally expensive two-dimensional problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' For our second experiment we consider a modified initial condition that contains a high-frequency component, namely 푢(푥, 푡 = 0) = 1 + 1 100 [cos(푥∕4) + cos(45푥∕4)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (20) The newly added perturbation does not fundamentally change the solution, but rather introduces a low-amplitude oscillation that persists throughout the temporal integration window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 3(b)-(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Moreover, the high-frequency mode does not make the computation more challenging for serial ERK methods, as evidenced by the convergence and efficiency plots that are nearly identical to those generated using the initial condition (19);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' compare figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 17 and 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We again consider the Parareal method from table 3 with either classical or repartitioned ERK integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 3(a) we show error as a function of the iteration number 푘, and see that repartitioning is again required to prevent instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' However, the high-frequency component has now prevented the repartitioned Parareal method from fully converging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='com/libpfasst/LibPFASST T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 7 of 33 Exponential Runge-Kutta Parareal (a) Error versus Iteration (b) Parallel Speedup versus Iteration (c) NLS Solution Figure 2: Error and speedup of the Parareal configuration from table 3 applied to the NLS equation (16) with the smooth initial condition (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Subfigure (a) shows the error at 푡final = 14 as a function of the Parareal iteration 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Line color is used to distinguish classical exponential integrators from repartitioned exponential integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' When 푘 = 0 the Parareal method is equivalent to running the coarse integrator with 푁푠∕푁푓 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Subfigure (b) shows the parallel speedup as a function of the iteration 푘;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' this compares the running time of the Parareal iteration to that of taking 푁푠 serial steps with the fine integrator 푓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Note that speedup is identical for both classical and repartitioned Parareal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Lastly, figure 2(c) shows the magnitude squared NLS solution |푢(푥, 푡)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (a) Error versus Iteration (b) Solution at 푡 = 14 (c) Magnified Solution at 푡 = 14 initial condition is (19) initial condition is (20) Figure 3: Subfigure (a) shows the error of the Parareal configuration from table 3 applied to the NLS equation (16) with the oscillatory initial condition (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Subfigure (b) compares the NLS solution at time 푡 = 14 for the two initial conditions (19) and (20) that are respectively drawn using a thick gray line and thin black line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The region enclosed by a blue square is magnified in subfigure (c) to highlight the small amplitude oscillation that arises from the initial condition (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' the method remains stable as 푘 increases, however the error does not improve beyond 3×10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This is our first indication that highly-oscillatory solutions will cause convergence problems for Parareal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Thus far, we have seen that repartitioning is essential for preventing instabilities in the exponential Parareal iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Therefore, we will no longer consider Parareal with classically partitioned ERK methods in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' However, we have also seen that repartitioning does not guarantee convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In [15, 64] it was shown that Parareal convergence on non-diffusive problems improves when the coarse propagator \ue233 more closely approximates the fine propagator \ue232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In our final motivating experiment, we will explore this phenomenon using an even more challenging initial condition that contains 45 spatial modes, namely 푢(푥, 푡 = 0) = 1 + 1 100 45 ∑ 푘=1 cos(푘푥∕4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (21) The NLS solution is now full of high-frequency information (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 20) that causes even serial ERK integrators to achieve slightly diminished accuracy for the same number of steps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' compare fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 19 to fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Based on the previous experiment we expect that the high-frequency oscillations will prevent the Parareal configuration in table 3 from rapidly T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 8 of 33 T 714010510X-1003 Iteratio4 n k56Fine Errort a 10 110 4 1 1Iclassicalrepartitioned10-10250 40 oeedup 30theoreticalachievedS 20 10 0 12 3 4 Iteration k5 63 Iteratio4 n k56Fine Errort a 10 110 4 1 1Iclassicalrepartitioned10-102-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='5 2 1-52V 0 XV 5100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='62 u 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='65-120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='X-11-10Exponential Runge-Kutta Parareal (a) Error versus Iteration (b) Parallel Speedup Figure 4: Parareal configuration with 푁푔 ∈ {1, 2, 3} (all other parameters are in table 3) applied to the nonlinear Schrödinger equation (16) with initial conditions (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Subfigure (a) shows how the error at 푦(푡final = 14) evolves as a function of the Parareal iteration 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Subfigure (b) shows how much faster the Parareal algorithms are compared to taking 푁푠 serial steps with the fine integrator 푓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Increasing 푁푔 improves convergence but also decreases parallel speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We again see significant decrease in speedup due to communication overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' converging to the fine solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We therefore change the accuracy of the coarse propagator \ue233 by considering two additional choices for the number of coarse steps, namely 푁푔 ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' By increasing 푁푔 (the number of steps of 푔 in \ue233) we can exchange parallel speedup for a more accurate coarse integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 4 we show error and parallel speedup as a function of the iteration number 푘 for these three Parareal configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' As expected, the Parareal configuration with 푁푔 = 1 does not converge to the fine solution within six iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' However, as 푁푔 increases we see that convergence properties improve significantly at the cost of decreased parallel speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In summary, repartitioning is required to prevent instabilities and the accuracy of the coarse integrator must be increased if we want to resolve high-frequency modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In the next section we will use linear stability analysis and linear convergence analysis to more carefully quantify these statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Linear stability and convergence analysis In this section we study the linear stability and convergence properties of exponential Parareal and provide a mathematical foundation for understanding the numerical experiments from section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Our analysis is based on the partitioned Dahlquist equation 푦′ = 휆1푦 + 휆2푦, 푦(0) = 1, (22) and follows closely with our previous works [16, 17] that respectively studied implicit-explicit Parareal and reparti- tioned exponential integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We note that for classical Parareal methods, there are many existing works studying stability and convergence [8, 69, 33, 64] including several that develop rigorous mathematical convergence bounds for diffusive problems [29, 67, 68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Unlike the more general nonlinear system (7), the partitioned Dahlquist equation provides a simple starting point for mathematical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This arises from the fact that any one-step exponential integrator, including Parareal, reduces to an iteration of the form 푦푛+1 = 푅(푧1, 푧2)푦푛 where 푧1 = ℎ휆1, 푧2 = ℎ휆2, (23) and ℎ is the method’s stepsize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Consequently, (22) is commonly used to study the stability of both exponential and implicit-explicit methods [5, 20, 53, 47, 17];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' in the case of exponential integrators the term 휆1푦 is exponentiated while the term 휆2푦 is treated explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' It should also be noted that (7) reduces to a system of decoupled, partitioned Dahlquist equations when the linear and nonlinear operator can be simultaneously diagonalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Since we are only considering non-diffusive equations, we assume that 휆1 and 휆2 are purely imaginary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The following table summarizes the relevant equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 9 of 33 50 40 oeedup 30theoreticalachievedS 20 ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 10 0 0 12 3 4 Iteration k5 63 teratio4 n k511 610~6Fine Errort a 4 10 or4 1 1I19 Ng= 310%Ng= 110-80 12 1Exponential Runge-Kutta Parareal Nonlinear system 퐲′ = 퐋퐲 + 푁(퐲) eig(퐋), eig( 휕퐍 휕퐲 ) ∈ 푖ℝ Partitioned Dahlquist 푦′ = 휆1푦 + 휆2푦 휆1, 휆2 ∈ 푖ℝ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' To estimate stability and convergence properties for a specific nonlinear system, we consider a family of partitioned Dahlquist equations with continuous 휆1, 휆2 values that respectively enclose the spectrums of the linear operator 퐋 and the Jacobian of the nonlinear operator 휕푁 휕퐲 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This rectangular parameter space in the scaled coordinates 푧1, 푧2 is 푍(ℎ) = {푧1 ∈ ℎ[− ̄휆1, ̄휆1], 푧2 ∈ ℎ[− ̄휆2, ̄휆2]} ̄휆1 = 푖휌(퐋), ̄휆2 = 푖 max 푡∈[푡0,푡final] 휌 ( 휕푁 휕퐲 (퐲(푡)) ) (24) where 휌(⋅) returns the spectral radius and ℎ is the stepsize required by the fine integrator to achieve a desired error tolerance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Ideally, we would like a Parareal configuration to be stable and rapidly convergent for any (푧1, 푧2) ∈ 푍(ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Finally, due to the limited stability of exponential integrators on non-diffusive equations, we must consider repartitioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' If one applies the repartitioning (13) to (15), the equations from the previous table have the following analogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Repartitioned nonlinear system 퐮′ = (퐋 + 휖퐃) ⏟⏞⏟⏞⏟ ̂퐋 퐮 + (푁(퐮) − 휖퐃퐮) ⏟⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏟ ̂푁(퐮) 퐋 = 퐔횲퐔−ퟏ, 퐃 = −퐔|횲|퐔−ퟏ Repartitioned Dahlquist 푦′ = (휆1 − 휖|휆1|) ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ ̂휆1 푦 + (휆2 + 휖|휆1|) ⏟⏞⏞⏞⏞⏟⏞⏞⏞⏞⏟ ̂휆2 푦 Since repartitioning preserves linearity, the iteration (23) for a repartitioned integrator simply becomes 푦푛+1 = 푅(푧1 + 휖|푧1|, 푧2 − 휖|푧1|)푦푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (25) The remainder of this section is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='1 we briefly quantify the parameter ranges that are pertinent for the discretized nonlinear Schrödinger equation from section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2 then contains simplified formulas for the Parareal method on the partitioned Dahlquist equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='4 we use linear analysis to study the stability and convergence properties of the exponential Parareal iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This allows us to quantify the stability effects of repartitioning, and to understand why high-frequency oscillations cause convergence problems for Parareal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='5 we then compare the predictions of linear analysis against the results of our nonlinear numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='6 we briefly analyze how certain Parareal parameters affect convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Lastly, we conclude with section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='7 where we discuss the implications of convergence analysis for the solution of partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Spectral radius of the nonlinear Schrödinger operators To analyze the numerical experiments from section 4, we first determine the parameters of the Dahlquist equation that most closely approximate the discretized nonlinear Schrödinger equation (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We proceed by bounding the spectral radius of the linear and nonlinear operators to estimate the rectangular parameter space 푍(ℎ) defined in (24): Linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The linear operator 퐋 for the discretized nonlinear Schrödinger equation with an even number of spatial grid points 푁푥 is 퐋 = diag(−푖퐤2) for 퐤 = 1 4[0:푁푥∕2 − 1, −푁푥∕2:−1]푇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (26) Using 푁푥 = 1024 and applying dealiasing we have ̄휆1 = 휌(퐋) = (341∕4)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' dealiasing removes the top one-third of the highest frequency modes so that only modes −341, … , 341 remain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Nonlinear operator Jacobian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Obtaining the exact spectral radius for the nonlinear Jacobian 휕푁 휕푢 is more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Instead, we estimate its magnitude by assuming there is no coupling between Fourier modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The continuous nonlinear operator in physical space is 2푖|푢|2푢, which when applied to a single mode 푢(푥) = 푎푘푒푖푘푥, leads to 2|푎푘|2푎푘푒푖푘푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Ignoring mode coupling, the discretized nonlinearity in Fourier space acts like the diagonal operator diag(2푖|퐮|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In each of the experiments from section 4, the elements of 퐮 are all bounded above by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='0001 throughout the temporal domain, so we estimate that ̄휆2 = max푡∈[0,14] 휌 ( 휕푁 휕퐲 (퐲(푡)) ) ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 10 of 33 Exponential Runge-Kutta Parareal Lastly, all the experiments from section 4 use a fine stepsize of ℎ = 14∕216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Therefore, using (24), the nonlinear Schrödinger equation can be approximately analyzed using the family of Dahlquist equations with scaled parameters 푧1 ∈ 푖[0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='6] and 푧2 ∈ 푖[−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='3] × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (27) To avoid imaginary numbers, it is convenient to consider the real-valued dimensions of this parameter region, namely 푟1 ∈ [0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='6] and 푟2 ∈ [−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='3] × 10−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (28) We will frequently refer back to these numbers as we study the stability and convergence properties of Parareal on the nonlinear Schrödinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Parareal for the partitioned Dahlquist equation We now present several formulas that describe the Parareal iteration (4) applied to the Dahlquist equation (22);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' these formulas were originally developed in [64] for unpartitioned linear problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We begin by considering the coarse and fine propagators (\ue233, \ue232) and their underlying integrators (푔, 푓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' When applied to (22) these methods reduce to the scalar iterations 푔: 푦푛+1 = 푅푔(푧1, 푧2)푦푛 \ue233: 푦푛+1 = 푅\ue233(푧1, 푧2)푦푛 = 푅푔(훿푧1, 훿푧2)푁푔푦푛 for 훿 = 푁푓 푁푔 푓: 푦푛+1 = 푅푓(푧1, 푧2)푦푛 \ue232: 푦푛+1 = 푅\ue232(푧1, 푧2)푦푛 = 푅푓(푧1, 푧2)푁푓 푦푛 (29) where ℎ is the stepsize and 푧1 = ℎ휆1 푧2 = ℎ휆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Note that for 푔 and 푓, 푦푛 corresponds to the 푛th fine step, while for \ue232 and \ue233, 푦푛 corresponds to the 푛th coarse step (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 1 for an illustration of coarse and fine steps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The Parareal iteration (4) then reduces to the matrix iteration 퐌\ue233퐲푘+1 = (퐌\ue233 − 퐌\ue232)퐲푘 + 퐛 (30) where the vector 퐲푘 = [푦푘 푗 ] ∈ ℝ푁푝+1 stores the solution at each coarse step, and the matrices 퐌\ue232, 퐌\ue232 ∈ ℝ푁푝+1,푁푝+1 and vector 퐛 ∈ ℝ푁푝+1 are 퐌\ue233 = ⎡ ⎢ ⎢ ⎢⎣ 퐼 −푅\ue233 퐼 ⋱ ⋱ −푅\ue233 퐼 ⎤ ⎥ ⎥ ⎥⎦ , 퐌\ue232 = ⎡ ⎢ ⎢ ⎢⎣ 퐼 −푅\ue232 퐼 ⋱ ⋱ −푅\ue232 퐼 ⎤ ⎥ ⎥ ⎥⎦ , 퐛 = ⎡ ⎢ ⎢ ⎢⎣ 푦0 0 ⋮ 0 ⎤ ⎥ ⎥ ⎥⎦ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (31) Note that the values 푅\ue233 and 푅\ue232 are the stability functions from (29) that depend on 푧1 and 푧2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Next, solving the recurrence relation (30) yields 퐲푘+1 = 푘 ∑ 푗=0 퐄푗퐌−1 \ue233 퐛, for 퐄 = 퐈 − 퐌−1 퐺 퐌−1 퐹 , (32) and we can now interpret the Parareal algorithm as a fixed point iteration that converges to the fine solution 퐲\ue232 = 푦0 [ 1, 푅\ue232, 푅2 \ue232, … , 푅 푁푝 \ue232 ]푇 ∈ ℝ푁푝+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (33) Lastly, if we define the error at the 푘th iteration as 퐞푘 = 퐲푘 − 퐲\ue232 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' the difference between the Parareal solution and the serial fine integrator solution), then the error at the 푘th iteration, evolves according to the matrix iteration 퐞푘 = 퐄퐞푘−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (34) To obtain (34), we substitute 퐲푘 = 퐞푘 + 퐲\ue232 into (30), left multiply by 푀−1 퐺 , then simplify using 퐌\ue232퐲\ue232 = 퐛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In the following subsections we will use (32) to study stability and (34) to study convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 11 of 33 Exponential Runge-Kutta Parareal 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Linear stability analysis Linear stability analysis [72, IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2] is a well-known technique that is used to determine the types of equations for which a time integrator is stable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' diffusive or advective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The analysis proceeds by considering the Dahlquist equation and determining the subset of parameters that lead to a stable iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' All one-step exponential integrators applied to the partitioned Dahlquist equation (22) reduce to the iteration (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The function 푅(푧1, 푧2) is the stability function of the method and its magnitude must be smaller than or equal to one to guarantee stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The stability region of a method contains all the (푧1, 푧2) pairs for which this holds true and is formally defined as 푆 = {(푧1, 푧2) ∈ ℂ2 ∶ |푅(푧1, 푧2)| ≤ 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (35) For a fixed set of parameters, Parareal is a one-step method that advances the solution by 푁푝 coarse timesteps, or equivalently, 푁푠 fine timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' If 푦푗 denotes the 푗th fine timestep, then Parareal applied to (22) reduces to the iteration 푦(푁푠(푛+1)) = 푅(푁푠푧1, 푁푠푧2)푦(푁푠푛) where 푧1 = ℎ휆1, 푧2 = ℎ휆2, (36) ℎ is the stepsize of the fine integrator 푓, and the stability function 푅 is 푅(푧1, 푧2) = 퐜2 ( 푘 ∑ 푗=0 퐄푗 ) 퐌−1 퐺 퐜1, 퐜ퟏ = [1, 0, … , 0]푇 ∈ ℝ푁푝+1, 퐜ퟐ = [0, … , 0, 1] ∈ ℝ푁푝+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (37) This stability function follows directly from (32);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 퐜1 is equivalent to 퐛 with 푦0 = 1 and 퐜2 extracts the solution at the final coarse step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We now apply linear stability analysis to study Parareal methods with classical and repartitioned ERK integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Since we are interested in non-diffusive problems with 휆1, 휆2 ∈ 푖ℝ, we only consider the two-dimensional stability region ̂푆 = {(푟1, 푟2) ∈ ℝ2 ∶ |푅(푖푟1, 푖푟2)| ≤ 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (38) Our aim is to determine if the Parareal method from table 3 is stable for the (푟1, 푟2) region (28) that encloses the eigenvalues of the discretized nonlinear Schrödinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 5 we compare the stability regions of the Parareal configuration from table 3 equipped with either classical or repartitioned ERK methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We immediately see that the stability regions associated with classical ERK integrators only encompass a small subset of the rectangular parameter region (28) and that the rate of instability worsens significantly as 푘 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In contrast, repartitioning greatly expands the stability region of the Parareal method and the remaining instabilities are sufficiently small that they will not affect the quality of the final solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Figures 21 and 22 from appendix D contain additional stability plots that reveal a wider range of 푟2 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Although repartitioning greatly improves stability, exponential Parareal is only stable when |푟2| ≪ |푟1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In other words, the linear term in (7) must contain the majority of the stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Overall, linear stability analysis is consistent with the convergence diagrams from figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 2 and 3 which show that Parareal with classical ERK methods grows increasingly unstable as the iteration count 푘 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' From the linear stability diagrams we also see that the Parareal iteration greatly magnifies the instabilities that are present in the serial ERK4 integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' For comparison, the serial ERK4 method is stable across the entire range of (푟1, 푟2) values shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 5 and its instability rates near the line 푟2 = 0 are smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='01;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 3 in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This implies that repartitioning is important for Parareal even on short timescales where serial exponential integrators do not require it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Linear convergence analysis We now study the convergence rate of the Parareal iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We again consider the partitioned Dahlquist equation (22) and determine the subset of parameters that lead to guaranteed rapid convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2 we showed that the difference between the Parareal solution and a serial fine integrator solution evolves according to the iteration (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Since Parareal fully converges after exactly 푁푝 iterations, the matrix 퐄 is nilpotent and the convergence rate cannot be derived from its spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Nevertheless, as originally proposed in [64], monotonic convergence is guaranteed if ‖퐄‖ < 1 since ‖퐞푘+1‖ ≤ ‖퐄‖‖퐞푘‖ < ‖퐞푘‖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (39) T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 12 of 33 Exponential Runge-Kutta Parareal Parareal Stability Regions and Instability Factors 푘 = 0 푘 = 2 푘 = 4 푘 = 6 Classical ERK Repartitioned ERK Figure 5: Stability regions for the Parareal configuration from table 3 with classical exponential integrators (top row) and repartitioned exponential integrators (bottom row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Each column corresponds to a different Parareal iteration 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The gray region is the stability region (38), and color shows the amplification factor |푅(푧1 = 푖푟1, 푧2 = 푖푟2)| outside the stability region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The 푟1 and 푟2 axis limits on the graphs correspond exactly to (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' For convergence to occur within a small number of Parareal iterations, we require ‖퐄‖ ≪ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' for example if ‖퐄‖ < 1∕10 it will take 10 iterations to reduce the error by 10 digits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In [15] we showed that the ∞-norm of 퐄 is ‖퐄‖∞ = 1 − |푅\ue233|푁푝 1 − |푅\ue233| |푅\ue233 − 푅\ue232|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (40) where the values 푅\ue233 and 푅\ue232 are the stability functions from (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The ∞-norm is convenient to use since it is both interpretable and easy to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Using (39) and (40) we define the convergence region \ue22f∞ to be the set of all (푧1, 푧2) pairs for which the ∞-norm of 퐄 is smaller than one \ue22f∞ = {(푧1, 푧2) ∈ ℂ ∶ ‖퐄(푧1, 푧2)‖∞ < 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (41) Since we are only considering non-diffusive equations with 휆1, 휆2 ∈ 푖ℝ, we will study the two-dimensional convergence region ̂\ue22f∞ = {(푟1, 푟2) ∈ ℝ ∶ ‖퐄(푖푟1, 푖푟2)‖∞ < 1} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (42) Note that the matrix 퐄 does not depend on the iteration 푘 so a single convergence region pertains to a Parareal configuration with an arbitrary 퐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We now apply linear convergence analysis to understand why high-frequency oscillations cause problems for Parareal and why increasing the number of coarse steps 푁푔 improves convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We again consider the three Parareal configurations from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 4 with 푁푔 ∈ {1, 2, 3} and all other parameters from table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We are primarily interested to see if the convergence regions of these three Parareal configurations enclose the rectangular region (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 6, we present the Parareal convergence regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' the red rectangles in each plot show the largest rectangular subset of (28) that can be enclosed inside each convergence region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The first observation is that the convergence regions near (푟1 = 0, 푟2 = 0) grow approximately linearly in size with respect to 푁푔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This observation follows directly from (40) since increasing 푁푔 makes the coarse propagator more accurate, therefore decreasing the quantity |푅\ue233−푅\ue232| (See remark 1 in appendix E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Overall, linear convergence analysis T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 13 of 33 2 0 2104 103 102X10 4 4 2107 106 105-4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='6 1101 100×10-4 4 2 2-2 4 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='8 r11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='62 0 2×10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='4 4 2-4 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='8 r11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='62 0 2×10 4 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='8 r11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='62 2X10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='4 4 2-4 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='8 r11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='6-24 2-4 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='8 r11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='6-2×10 4 2-4 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='8 r11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='6-2×10 4 2-4 h 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='8 r11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='6-2×10 4 2-4 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='8 r11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='6Exponential Runge-Kutta Parareal 퐍퐠 = ퟏ 퐍퐠 = ퟐ 퐍퐠 = ퟑ Magnified 푟2 axis Magnified 푟2 axis Magnified 푟2 axis Figure 6: Convergence regions (42) for the three Parareal configurations from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The columns represent different choices for 푁푔 and the bottom row shows a magnified 푟2 axis compared with the top row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Color represents the ∞-norm of the matrix 퐄, from (32), and the red rectangles are the largest rectangular subset of the 푟1 and 푟2 range from (28) that can be contained inside the convergence region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The 푟1-coordinate of the labeled point corresponds to the width of the rectangular subset rounded to three digits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' it is defined as 푟max 1 = max휌 subject to {푟1 = 휌, 푧2 ∈ [−2ℎ, 2ℎ]} ⊆ ̂퐶∞ for ℎ = 14∕216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' confirms that increasing 푁푔 leads to a Parareal configuration that will resolve a larger number of high-frequency temporal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The second observation is that the convergence regions are small and fail to fully enclose (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' More precisely, while all of the 푟2 range is inside the convergence region, less than twenty percent of the 푟1 range is included, even when 푁푔 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' However, recall that in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 4 we saw that the Parareal configuration with 푁푔 = 3 was able to accurately converge to the fine solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' we will explore this fact in more detail in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Our third and final observation involves convergence rates for small, fixed (푟1, 푟2) and follows directly from remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Specifically, if we let 푞 be the order of the coarse integrator, then, for fixed (푟1, 푟2), ‖퐄‖∞ = \ue23b(1∕푁푔 푞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Therefore increasing 푁푔 will also increase the Parareal convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Validating convergence results for the nonlinear Schrödinger equation We now validate how closely the predictions of linear convergence analysis, made using the Dahlquist parameters (27), align with the results from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The nonlinear Schrödinger equation was spatially discretized using a Fourier spectral discretization that represents the solution as the sum of 푀 Fourier modes, such that 푢(푥, 푡) = 푀∕2−1 ∑ 푛=−푀∕2, 퐚푛(푡)푒푖푘푥∕4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (43) T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 14 of 33 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='320 (0188,00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='05-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='05 01 r10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='073, 0510-45 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='1 r10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='133, 0510-45 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='1 r10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='3200188, 0)510-45 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='Exponential Runge-Kutta Parareal 푁푔 푟max 1 Convergent 퐚푛(푡) 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='073 푛 ∈ [−73, 73] 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='133 푛 ∈ [−99, 99] 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='188 푛 ∈ [−118, 118] Figure 7: Monotonically convergent Fourier coefficients, as predicted by linear convergence analysis, for the three Parareal configurations from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 4 with 푁푔 ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The set of convergent mode indices is defined as {푛 ∈ ℤ ∶ 푟1(푛) ≤ 푟max 1 } for 푟1(푛) = ℎ푛2∕16 and ℎ = 14∕216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The table contains 푟푚푎푥 1 values (the 푥 coordinates of the labeled points in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 6) along with the indices of the corresponding convergent coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The blue line in the plot shows 푟1(푛), the solid gray horizontal lines correspond to the three 푟max 1 values, and the pairs of vertical dashed lines are the upper and lower bounds for the predicted convergent coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The Fourier coefficients 퐚푛(푡) evolve according to (7) and (17) with 퐲 = [퐚0, … , 퐚푀∕2−1, 퐚−푀∕2, … , 퐚−1]푇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Therefore, the differential equation that governs the 푛th coefficient is ̇퐚푛 = 휆1(푛)퐚푛 + [푁(퐲)] 훼(푛)−푀∕2 where 휆1(푛) = 푖푛2∕16, (44) [푁(퐲)]푗 is the 푗th component of the nonlinearity, and 훼(푛) = [1 + 푛 + 푀∕2] (mod 푁).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' To conduct linear analysis, we replace the coupled nonlinearity with the decoupled linear term 휆2퐚푛, with 휆2 ∈ 푖[−2, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' It then follows that a Parareal iteration with fine timestep ℎ will monotonically converge to the solution of 퐚푛(푡) only if (ℎ휆1(푛), ℎ휆2) is inside the convergence region ̂퐶∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In other words, linear convergence analysis predicts that Parareal will only monotonically converge to the subset of Fourier coefficients 퐚푛(푡) for which 푛 satisfies the set inequality {푟1 = 푟1(푛), 푟2 ∈ [−2ℎ, 2ℎ]} ⊆ ̂\ue22f∞ for 푟1(푛) = ℎ푛2∕16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (45) Ignoring stability considerations, all other 퐚푛(푡) will remain at the accuracy achieved by the coarse integrator until 푘 → 푁푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We can simplify the condition (45) by introducing 푟max 1 = max 휌 subject to {푟1 = 휌, 푟2 ∈ [−2ℎ, 2ℎ]} ⊆ ̂\ue22f∞, (46) which is the width of the largest rectangle that includes the entire 푟2 range and is enclosed by the convergence region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The values of 푟max 1 for the three Parareal configurations considered in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 4 are the 푟1-coordinates of the labeled points in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Using (46) it follows immediately that the condition (45) is equivalent to the inequality 푟1(푛) < 푟max 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (47) In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 7 we present a table of the 푟max 1 values for the three parareal configurations from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 4, along with the resulting estimates of the monotonically convergent Fourier modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Then in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 8 we validate these estimates by comparing the Fourier coefficients 퐚푛(푡) obtained using the Parareal iteration to those obtained using the serial ERK4 integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Overall we see that linear convergence analysis very accurately predicts the convergent Fourier coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Moreover, we see that the accuracy of all Fourier coefficients with an 푟1(푛) that was outside of the convergence region did not improve substantially beyond what was achieved using the coarse integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Convergence regions for additional Parareal configurations Convergence regions depend on all the Parareal parameters from table 2 and on the repartitioning constant 휌 from (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Here we investigate the effects of changing the coarse integrator \ue233 and the repartitioning constant 휌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Figure 9 presents convergence regions for Parareal configurations with \ue233 ∈ {ERK1, ERK2, ERK3, ERK4} and all other parameters taken from table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We see that using a higher-order coarse integrator results in a larger convergence region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This follows directly from (40) since the increased accuracy of a high-order coarse propagator decreases the T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 15 of 33 0 ficient inoI 73 lex n99 1180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='073- --0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='133- -- --- ■0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='188-H- L- L 中 -- ri(n- + I 118-99I 73 COeExponential Runge-Kutta Parareal 퐍퐠 = ퟏ 퐍퐠 = ퟐ 퐍퐠 = ퟑ Error in Fourier Coefficient 퐚푛(14) Error in Fourier Coefficient 퐚푛(14) Error in Fourier Coefficient 퐚푛(14) Error Norm vs Iteration Error Norm vs Iteration Error Norm vs Iteration Coarse (퐾 = 0) Parareal (1 ≤ 퐾 ≤ 5) Parareal (퐾 = 6) Fine Propagator (퐾 = 푁푝) Figure 8: Plots describing the same numerical experiment as the one from fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Each column corresponds to a Parareal configuration with a different value of 푁푔 and the colors represent different Parareal iteration numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Top Row: Error in the Fourier coefficients 퐚푛(푡) of the solution (43) at 푡 = 14 for Parareal with 퐾 ∈ {0, … , 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The black line corresponds to the fine propagator \ue232 run in serial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The two vertical dotted lines in each plot are the upper and lower bounds for the convergent spatial modes as predicted by linear analysis and are identical to those shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Bottom Row: Error norm between the reference solution and the Parareal method at 푡 = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' These plots are identical to the one shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' quantity |\ue233 − \ue232|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Therefore, replacing a low-order coarse propagator with a higher-order one, provides another way to improve convergence for high-frequency temporal modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Naturally, any convergence gains must be weighed against the decrease in parallel speedup (5) caused by the more expensive high-order coarse propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Next, we briefly discuss how the repartitioning parameter 휌 from (15) affects convergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' recall that 휌 is the angle (in radians) that the eigenvalues of the linear operator are rotated into the left-half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Figure 10 contains convergence regions for the Parareal configuration from table 3 with repartitioning parameters 휌 ∈ {0, 휋∕256, 휋∕64, 휋∕16}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' When no repartitioning is applied (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 휌 = 0) an exponential integrator will exactly solve a Dahlquist equation (22) with 휆2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Therefore, the Parareal convergence region for 휌 = 0 extends infinitely along the line 푟2 = Im(ℎ휆2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Any amount of repartitioning destroys the exactness of the integrator along this line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In practice this is not important since imposing 휆2 = 0 is equivalent to forcing the nonlinearity 푁(푡, 푦) in (2) to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' It should be noted however, that increasing the repartitioning parameter leads to a more subtle contraction of the overall convergence region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Therefore, to maximize convergence for high-frequency information, one should select the smallest repartitioning constant that ensures stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Implications of convergence analysis for solving non-diffusive PDEs As demonstrated by linear analysis, Parareal only converges rapidly to equation components that are not overly oscillatory in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' There are many factors affecting the number of high-frequency temporal modes present in a spatially discretized partial differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' High-order dispersive derivatives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 푢푥푥푥 or 푖푢푥푥) possess continuous T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 16 of 33 0 icient- : 73 index n1701010° JO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' n- --10-1- 170-73 coeff0 icientindex 99 n1701010° JO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' n-- -- 10- 1170-99 coeff0 icientindex 118 n1701010° JO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' n- -- -- h- 10-1170 -11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='8 coeffK士10Err 10Fine SollutionError10 r 01-一K士10Err 10Fine Sollution Err10 r 01-一K士10Err 10Fine Solljtion kror10 r 01-一Exponential Runge-Kutta Parareal \ue233 = ERK1 \ue233 = ERK2 \ue233 = ERK3 \ue233 = ERK4 Figure 9: Convergence regions for the Parareal configuration from table 3 with different coarse integrators \ue233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 휌 = 0 휌 = 휋∕256 휌 = 휋∕64 휌 = 휋∕16 Figure 10: Convergence regions of the Parareal configuration from table 3 with four repartitioning constants 휌 from (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' spectrums with large imaginary eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The extent to which the continuous spectrum causes convergence problems will depend on the choice of spatial discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Non-diffusive discretizations, such as Fourier pseudo- spectral methods, will present the greatest challenge since the discretized linear operators will also have purely imaginary eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Using fine spatial grids will also increase the total number of high-frequency components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Finally, there is the question of whether high-frequency information is necessary for obtaining accurate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This will depend on the initial condition, the length of the integration window, and the characteristics of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' PDEs or initial conditions that cause rapid spectral broadening within the integration window will be the most challenging to solve using Parareal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In summary, limited convergence for high-frequency temporal components will manifest as inaccuracies in high-frequency spatial modes of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' For the generic initial value problem (7), we can extend the linear analysis developed in this section to estimate the convergence properties of a given Parareal configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We start by making the following two assumptions: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The linear operator 퐋 is diagonalizable and (휆푛, 퐯푛) are the 푛th eigenvalue and eigenvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This allows us to express the solution as a linear combination of the eigenvectors, 퐲(푡) = ∑푀 푛=0 푎푛(푡)퐯푗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The linear operator contains the majority of the stiffness such that 휌(퐋) ≫ 휌( 휕푁 휕푦 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We then bound the spectrum of the linear and nonlinear operators and define the region \ue23e(ℎ) = {푟1 ∈ ℎ[0, 푐1], 푟2 ∈ ℎ[−푐2, 푐2]} 푐1 = 휌(퐋), 푐2 = max 푡 휌 ( 휕푁 휕퐲 (퐲(푡)) ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (48) To proceed we must first ensure that a given Parareal configuration is sufficiently stable for all (푟1, 푟2) in \ue23e(ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' If this holds true, we then determine the largest rectangular subset of \ue23e(ℎ) that: (i) is enclosed by the Parareal convergence region (42) and (ii) contains the entire 푟2 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The width of this region is 푟max 1 (ℎ) = max 휔 subject to {푟1 = 휔, 푟2 ∈ ℎ[−푐2, 푐2]} ⊆ ̂\ue22f∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (49) The value of 푟max 1 (ℎ) will depend on all Parareal parameters except for 퐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Finally, we estimate that a Parareal iteration with fine stepsize ℎ will monotonically converge to the fine integrator solution of any coefficient 푎푛(푡) where 푛 satisfies |ℎ휆푛| < 푟max 1 (ℎ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (50) T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 17 of 33 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='04 r10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='00 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 ~8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='04 r10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='00 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 ~8-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='0 r140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='08200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='0 r140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='08200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='0 r140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='08200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='0 r140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='08200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03~-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='0 r140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='08200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='0 r140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='08200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='0 r140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='08200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='0 r140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='08200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='03 0Exponential Runge-Kutta Parareal All the remaining coefficients will retain the accuracy achieved with the coarse integrator and fail to converge for small iteration count 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Although these estimates are rooted in linear theory, our results in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='5 demonstrate that this approach has the potential to accurately predict convergence for nonlinear problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Higher-dimensional numerical experiments We now demonstrate that it is possible to achieve meaningful parallel speedup using an exponential Parareal method on higher-dimensional non-diffusive equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We conduct two additional numerical experiments in which we solve the dispersive Kadomtsev-Petviashvili (KP) equation and the hyperbolic Vlasov-Poisson (VP) equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Both PDEs are equipped with periodic boundary conditions and discretized in space using a Fourier spectral method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Since analytical solutions are not known, we compute a reference solution using ERK4 with a very small timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The error is then defined as ‖퐲ref − 퐲method‖∞∕‖퐲ref‖∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (51) where 퐲 represents the solution in physical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Below we describe the equations, their initial conditions, and the corresponding numerical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The Kadomtsev-Petviashvili (KP) equation is (푢푡 + 6푢푢푥 + 푢푥푥푥 ) 푥 + 3휎2푢푦푦 = 0 (52) where 휎2 = −1 leads to KPI that models thin films with large surface tension, while 휎2 = 1 leads to KPII that models water waves with small surface tension [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Both equations admit the soliton solution 푢(푥, 푦, 푡) = 2푝2sech(푝(푥−4푝2푡)) where 푝 is a free variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The stability of a perturbed soliton depends on the sign of 휎2, with the KPI solution being unstable and the KPII solution being stable [28, 45, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' For any well-localized solution in 푥, the KP equation can be expressed in evolution form as 푢푡 + 6푢푢푥 + 푢푥푥푥 + 3휎2휕−1푢푦푦 = 0 where 휕−1푓 = 1 2 [ ∫ 푥 −∞ 푓(푠)푑푠 − ∫ ∞ 푥 푓(푠)푑푠 ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (53) To ensure smoothness in time, the initial condition must satisfy the following equality at 푡 = 푡0 ∫ ∞ −∞ 푢푦푦(푥, 푦, 푡)푑푥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (54) Any initial condition that does not satisfy this constraint will produce a solution that satisfies (54) for 푡 > 푡0, leading to a temporal discontinuity at 푡0 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' For our numerical experiment we consider the KPI equation equipped with periodic boundary conditions on the domain 푥 ∈ [−8휋, 8휋], 푦 ∈ [0, 8휋].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We spatially discretize using 972 grid points in 푥 and 750 grid points in 푦, and dealias using the standard 3/2 rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We integrate the equation in Fourier space where the operator 휕−1 is equivalent to the Fourier multiplier −푖∕푘푥;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Note that when 푘푥 = 0 this mode is singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' However, for any initial condition that satisfies (54) we can simply set this multiplier to zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' for more general initial conditions, numerical regularization must be added [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' As in [45] we select our initial condition to be a soliton with a perturbed phase 푢(푥, 푦, 푡 = 0) = 2sech2 ((푥 + 4휋) + 훿 cos(푦∕4)) , 훿 = 1∕5, (55) and integrate the equation to time 푡final = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Our initial condition satisfies (54), therefore, no regularization is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' As shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 11, the perturbation is unstable and leads to the formation of a two-dimensional soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The hyperbolic Vlasov-Poisson (VP) equation is 푓푡 + 푣푓푥 + 퐸(푥, 푡)푓푣 = 0, for 퐸푥(푥, 푡) = −1 + ∫ ∞ −∞ 푓(푥, 푣, 푡)푑푣, (56) and describes the evolution of charged particles in an electric field [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Our numerical experiment is based on the bump-on-tail experiment from [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Specifically, we equip the VP equation with periodic boundary conditions on the T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 18 of 33 Exponential Runge-Kutta Parareal Figure 11: Solution of the KPI equation (53) at 푡 = 4 with initial conditions (55).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' domain 푥 ∈ [0, 20휋], 푣 ∈ [−8, 8], and spatially discretize using a 1024 point Fourier discretization in both 푥 and 푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Our initial condition is 푓(푥, 푣, 푡 = 0) = ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='9 √ 2휋 푒−푣2∕2 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2 √ 2휋 푒−2(푣−4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='5)2 ) ( 1 + 4 100 cos(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='3푥) ) (57) and the solution is integrated to time 푡final = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' To preserve a diagonal linear operator we solve the equation in physical 푣 space and Fourier 푥 space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' see (67) in appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' As shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 12 the bump-on-tail initial condition excites modes that lead to complex dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Figure 12: Solution of the Vlasov-Poisson equation (56) at 푡 = 50 for the initial condition (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Parareal parameter selection and experiment overview The Parareal configurations we selected to solve the KP and VP equations are described in table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' For both equations, we considered multiple configurations that differ only in the number of coarse steps 푁푔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We vary this parameter to demonstrate the improved convergence properties associated with larger 푁푔 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' For the fine integrator 푓 we always selected ERK4 and set the total number of steps 푁푠 so that a fully-converged Parareal method produces a T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 19 of 33 0 10 c1020 1020H04 n10 5y 0-20660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 2 0 2 8/ 7 6 4240 505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='330200 1002 7 4 aExponential Runge-Kutta Parareal (a) Kadomtsev-Petvaishvili (KP) Parareal Parameters 푁푝 8192 푁푓 32 푓 ERK3 퐾 1, … , 28 푁푠 218 푁푔 {1, 2, 3} 푔 ERK4 푟max 1 values 푁푔 = 1 푁푔 = 2 푁푔 = 3 푟max 1 (ℎ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='0642 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='155 (b) Vlasov-Poisson (VP) Parareal Parameters 푁푝 2048 푁푓 64 푓 ERK4 퐾 1, … , 12 푁푠 217 푁푔 {2, 3} 푔 ERK3 푟max 1 values 푁푔 = 2 푁푔 = 3 푟max 1 (ℎ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='102 Table 4 Parareal parameters used to solve the KP equation (53) with initial conditions (55) and the Vlasov-Poisson equation (56) with initial conditions (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In the right-most tables we present the associated 푟max 1 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' All 푟max 1 values are computed using the stepsize ℎ = 푡final∕푁푠, the repartitioning parameter 휌 = 휋∕128, and by assuming that the stiffness in the nonlinear term is negligible such that 푐2 = 1 in (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' highly accurate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The remaining parameters were then determined by balancing the achievable parallel speedup with the size of the convergence regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The results for the numerical experiments in this section will be summarized in three plots: (i) error versus iteration 퐾, (ii) parallel speedup versus iteration 퐾, and (iii) error versus run-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In the error versus run-time plots we will compare the efficiency of the Parareal iteration to that of the coarse and fine ERK integrators run in serial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' All experiments were performed using 32-core Haswell nodes on the Cray XC40 Cori at the National Energy Research Scientific Computing Center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' For the VP equation we collected timing results by running the Parareal iteration on 64 nodes (2048 compute cores).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The full KP experiment requires 256 nodes (8096 compute cores) which exceeded our available computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Therefore, we ran the Parareal iteration in serial on a single node to determine the convergence curves, and then extrapolated the achievable speedup from a smaller experiment where we solved the KP equation on the shortened interval 푡 ∈ [0, 1] using 64 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Lastly, for each Parareal configuration we compute the convergent spatial modes as predicted by the linear stability analysis from section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' For simplicity, we assume that any stiffness in the nonlinear term is negligible so that convergence depends exclusively on the eigenvalues of the linear operator (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 푐2 = 0 in (48)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Kadomtsev-Petvaishvili – results and discussion The KP equation is challenging to solve because the third-order derivative term leads to a linear operator with large imaginary eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' As we have seen in sections 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='5 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='7, these eigenvalues determine the degree of temporal oscillation present in the spatial Fourier coefficients of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Therefore, we must select a Parareal configuration whose convergence region contains at least a large subset of these eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Our proposed configurations are described in table 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The choice of total steps 푁푠 ensures that a fully-converged Parareal iteration will produce a solution with an accuracy of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2 × 10−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Moreover, we selected ERK3 as the coarse integrator because it is stable at sufficiently large stepsizes (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 23) and the convergence region for an ERK3, ERK4 pairing is larger than that of an ERK2, ERK4 pairing (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We can estimate the convergent spatial modes for each choice of 푁푔 using the stability analysis developed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The eigenvalues of the linear operator 퐋 are 퐋(푘푥, 푘푦) = { −푖(휔푥푘푥)3 푘푥 = 0 −푖(휔푥푘푥)3 + 푖 (휔푦푘푦)2 휔푘푥 푘푥 ≠ 0 (58) where 푘푥 and 푘푦 are integer Fourier wavenumbers and 휔푥 = 2휋∕퐿푥 = 1∕8, 휔푦 = 2휋∕퐿푦 = 1∕4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The convergent spatial modes lie inside the region \ue22d = {(푘푥, 푘푦) ∈ ℤ2 ∶ ℎ|퐋(푘푥, 푘푦)| < 푟max 1 (ℎ)} for ℎ = 2−16, (59) T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 20 of 33 Exponential Runge-Kutta Parareal where the values of 푟max 1 depend on 푁푔 and are contained in table 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 13 we overlay the region \ue22d onto the linear operator 퐋 and the Fourier transformed final solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This allows us to estimate which spatial modes will be accurately computed by each Parareal configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Although none of the regions \ue22d enclose the entire (푘푥, 푘푦) domain, the coefficients for the highest-frequency spatial modes are small and only need to be resolved if we require an extremely accurate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (a) Magnitude of the scaled KP linear operator eigenvalues (b) KP Solution at 푡 = 8 (Fourier transformed in 푥,푦) Figure 13: Magnitude of the linear operator eigenvalues (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 13a) and final solution in Fourier space (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 13b) for the KP equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The axis in both plots are the Fourier wavenumbers in 푥 and 푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The black contours shows the boundaries of the region \ue22d from (60) that encloses all convergent spatial modes as predicted by linear convergence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The dotted, dashed, and solid line respectively correspond to the region \ue22d for Parareal configurations with 푁푔 = 1, 푁푔 = 2, and 푁푔 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' From linear analysis, we expect that each Parareal iteration will converge monotonically for all spatial modes inside its corresponding region \ue22d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 14 we show convergence, speedup, and efficiency results for the exponential Parareal methods applied to the KP equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We divide our discussion of the results into three parts: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' No exponential Parareal method converged monotonically across the entire 푘 range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Instead they exhibited a rapid reduction in error during the first few iterations before entering a plateau of slow convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This behavior is expected, since none of the Parareal convergence regions enclose the full spectrum of the discretized KP linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' As predicted by linear analysis, increasing 푁푔 improves convergence, and the Parareal configuration with 푁푔 = 3 is able to obtain a solution that is comparable in accuracy to the serial fine integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In contrast, the Parareal configurations with 푁푔 ∈ {1, 2} do not resolve a sufficient number of high-frequency spatial modes to achieve fine error within 27 iterations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' nevertheless, both methods improve the coarse solution by multiple orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' These results are analogous to those for the NLS equation shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Parallel speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In contrast to the one-dimensional NLS equation, the KP equation is more computationally expensive to integrate over a single timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This is due to the fact that a right-hand-side evaluation now requires multiple two-dimensional discrete Fourier transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This reduces the significance of communication costs, and we now see very good agreement between the theoretical and achieved parallel speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In summary, though the penalties incurred due to communication costs are mildly visible (notice that all dashed lines are slightly below the solid lines in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 14b), the theoretical speedup estimate (5) provides a realistic measure for real-world performance of the Parareal iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Perhaps the most important result is the error versus time plot, from which we see that all three Parareal configurations are able to compute high-accuracy solutions significantly faster than the serial ERK methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 21 of 33 00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='51-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='51Ng=1 ---- Ng=2 T— Ng = 3-1 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='5100 2 100 300-200K I 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2 2 h T010100 0F 90 6 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='4200-300-200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='00 0100 200 300-200)nl) 0g10( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='10100 04) 5 - 0200-300-200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='00 0Exponential Runge-Kutta Parareal (a) Error versus Iteration (b) Parallel Speedup versus Iteration (c) Error versus Time Figure 14: Numerical results for the KP equation using the three Parareal configurations from table 4, that are differentiated with different colored lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (a) Solution error at the final time 푡 = 4 as a function of Parareal iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (b) Theoretical speedup (5) and achieved speedup from the numerical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (c) Efficiency diagram comparing the Parareal configurations to the serial coarse and fine integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Moreover, despite their failure to converge to the fine solution within 28 iterations, the Parareal configurations with 푁푔 ∈ {1, 2} remain the fastest methods for obtaining moderately less accurate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We summarize this in the table below: 푁푔 Error Tol at 푘 = 28 Speedup compared to serial ERK4 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='7 × 10−7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='09x 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='3 × 10−8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='15x 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='8 × 10−9 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='71x Lastly we note that fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 14b shows both the efficiency for the Parareal method using the theoretical speedup (dashed lines) from (5) and the achieved speedup (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Unlike our one-dimensional experiments, we now see good agreement between these two qualities and the corresponding lines look nearly identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Vlasov-Poisson – results and discussion The Vlasov-Poisson equation does not contain high-order spatial derivatives, therefore it is possible to select Parareal parameters that simultaneously offer good Parallel speedup and convergence properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Our proposed Parareal configurations are described in table 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The choice of total steps 푁푠 ensures that a fully-converged Parareal iteration will produce a solution with an accuracy of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='6 × 10−8 (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We again apply linear analysis to estimate the convergent spatial modes for each choice of 푁푔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The eigenvalues of the linear operator 퐋 are 퐋(푘푥, 푣) = 푖휔푥푘푥푣 for 푘푥 ∈ ℤ, 푣 ∈ \ue242, and \ue242 = {−8 + 16푗∕푁푣}푁푣−1 푗=0 (60) where 푣 represents a discrete grid point on the domain [−8, 8], 푘푥 is the Fourier wavenumber in 푥, and 휔푥 = 2휋∕퐿푥 = 1∕10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The convergent spatial modes lie inside the region \ue22e = {(푘푥, 푣) ∈ ℤ × \ue242 ∶ ℎ|퐋(푘푥, 푣)| < 푟max 1 (ℎ)} for ℎ = 50∕217, (61) where the values of 푟max 1 are contained in table 4b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Because we are only Fourier transforming in the 푥 direction, it is important that our Parareal configuration accurately computes all the components in the 푣 domain since we cannot assume spectral decay in the physical 푣 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 15 we overlay the convergence region \ue22e onto the linear operator 퐋 and the transformed final solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The convergence regions for Parareal configurations with both 푁푔 = 1 and 푁푔 = 2 enclose the entire discrete (푘푥, 푣).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 22 of 33 teratioFine Errora rror 10 64 3 1I t19 Ng= 3100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='N10F 2eratiol1 k2%10Speed 30 S 2040 up50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=" theoreticalachievedF 2上 6 10 I11) ive Tir10 ne (sec21 a rror 6 10'4 1I t100ERK3 ERK410 10°1( RelatExponential Runge-Kutta Parareal note that the largest diagonal element of the scaled linear operator ℎ퐋 is 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='0693, therefore the convergence region for the Parareal configuration with 푁푔 = 1 just barely encloses the eigenvalues since 푟max 1 (ℎ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='0740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 16 we show convergence, speedup, and efficiency plots for the exponential Parareal method applied to the VP equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We again divide our discussion of the results into three parts: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The Parareal method with 푁푔 = 2 failed to converge, while the method with 푁푔 = 3 displayed monotonic convergence and achieved the fine error tolerance after eight iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The failure of convergence for 푁푔 = 2 is likely due to several reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' First, linear analysis is not guaranteed to provide an accurate prediction for all nonlinear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Moreover, linear analysis predicts that the Parareal with 푁푔 = 2 is only just barely convergent, so a larger safety margin may be required to properly predict convergence on nonlinear problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Lastly, it is also possible that the divergent iteration is due to instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Specifically, our assumption that the nonlinear term is completely non-stiff may be inaccurate due to the presence of the term −푓푣 in the nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Fortunately, modestly increasing 푁푔 resolves these issues and leads to a stable, monotonically convergent Parareal iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Parallel Speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' As with the KP equation, we see very good agreement between the theoretical and achieved parallel speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We can again see very minor penalties due to communication (notice that all dashed lines are slightly below the solid lines), however the differences are even smaller than those for the KP equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This follows from the fact that the cost per timestep is more expensive for the VP equation than for the KP equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The Parareal configuration with 푁푔 = 3 was able to obtain a solution 24 times faster than the serial ERK4 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 푁푔 Error Tol at 푘 = 8 Speedup compared to serial ERK4 3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='6 × 10−8 24x This substantial gain in speedup compared to the KP experiment is made possible by the lack of high-oscillatory temporal components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' This allowed us to run the coarse integrator at a significantly larger stepsize relative to the fine integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' For comparison, the coarse integrator of the Parareal configurations with 푁푔 = 3 for the KP and VP equations, were respectively run with a stepsize that was 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='6 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='3 times larger than the fine integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Overall this experiment demonstrates the potential for very significant speedup when applying exponential Parareal to accurately solve hyperbolic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Conclusions and future work In this paper we applied exponential integrators within the Parareal iteration and presented linear analysis that can be used to study stability and convergence properties of the resulting methods on non-diffusive equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We then demonstrated the ability of exponential Parareal methods to achieve significant parallel speedup on multiple non- diffusive partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We draw two main conclusions from this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' First we showed that repartitioning is essential for obtaining a Parareal configuration that is stable on stiff non-diffusive equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Second, through linear analysis we were able to better understand the convergence characteristics of the Parareal iteration in the absence of diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Specifically we saw that the Parareal iteration achieves fine integrator accuracy for low-frequency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' non-stiff) oscillatory modes and coarse integrator accuracy for high-frequency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' stiff) oscillatory modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' When solving non-diffusive partial differential equations this phenomenon makes it impossible to guarantee rapid convergence for high-frequency spatial modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Therefore, exponential Parareal is best suited for non-diffusive equations and initial conditions that do not cause rapid spectral broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' To the best of the authors’ knowledge, this is the first paper to investigate the usage of exponential integrators within a parallel-in-time framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Our initial results look promising as we have demonstrated the ability to achieve parallel speedup using exponential Parareal on both hyperbolic and dispersive equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Nevertheless, there are still many avenues that require further exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In particular all of the numerical experiments presented in this paper involve diagonal linear operators that greatly simplify the computation of the exponential 휑-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In future work we plan to study exponential Parareal integrators in the more general setting with non-diagonal linear operators and examine the resulting effects on computational performance and parallel speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 23 of 33 Exponential Runge-Kutta Parareal (a) Magnitude of the VP linear operator eigenvalues (b) VP Solution at 푡 = 50 (Fourier transformed in 푥 only) Figure 15: Magnitude of the linear operator eigenvalues (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 15a) and the transformed final solution (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 15b) for the Vlasov-Poisson equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The axes in both plots represent the Fourier wavenumber in 푥 and the spatial variable 푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The region \ue22e from (61) for both the Parareal configurations from table 4 encloses the entire discrete (푘푥, 푣) domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' therefore no black contours are shown on the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (a) Error versus Iteration (b) Parallel Speedup (c) Error versus Time Figure 16: Numerical results for the VP equation using the two Parareal configurations from table 4, that are differentiated with different colored lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (a) Solution error at the final time 푡 = 50 as a function of Parareal iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (b) Theoretical speedup (5) and achieved speedup from the numerical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (c) Efficiency diagram comparing the Parareal configurations to the serial coarse and fine integrators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Acknowledgements The work of Buvoli was funded by the National Science Foundation, Computational Mathematics Program DMS- 2012875.' metadata={'source': 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Solving ordinary differential equations II, Springer Berlin Heidelberg, 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 26 of 33 Exponential Runge-Kutta Parareal A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Method coefficients This appendix contains the Tableaus described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='1 for the exponential Runge-Kutta integrators that are used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We use the abbreviation 휑푖,푗 = 휑푖(푐푗ℎ퐋) for the 휑-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' ERK1: first-order exponential Euler method (10) 0 휑1 ERK2: second-order method from [20] 0 1 휑1 휑1 − 휑2 휑2 ERK3: third-order method from [20] 0 1 2 1 2휑1,2 1 −휑1 2휑1 휑1 − 3휑2 + 4휑3 4휑2 − 8휑3 −휑2 + 4휑3 ERK4: fourth-order method from [53] 0 1 2 1 2휑1,2 1 2 1 2휑1,2 − 휑2,2 휑2,2 1 휑1 − 2휑2 0 2휑2 휑1 − 3휑2 + 4휑3 2휑2 − 4휑3 2휑2 − 4휑3 −휑2 + 4휑3 T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 27 of 33 Exponential Runge-Kutta Parareal Error versus Stepsize – Initial Condition (19) Error versus Computational Time – Initial Condition (19) Figure 17: Convergence diagram (left) and precision diagram (right) for the exponential Runge-Kutta methods listed in appendix A run on the nonlinear Schrödinger equation (16) with initial condition (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Colored lines correspond to repartitioned integrators (rERK) and gray lines correspond to unmodified exponential integrators (ERK) – repartitioning has no effect on this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The black crosses on the ERK3 and ERK4 method respectively correspond to the stepsizes of the coarse and fine integrators for the Parareal method described in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Error versus Stepsize – Initial Condition (20) Error versus Computational Time – Initial Condition (20) Figure 18: Identical to fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 17 except we are now considering the initial condition (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Nonlinear Schrödinger Serial ERK Results We solve the nonlinear Schrödinger equation (16) using the serial ERK methods from from appendix A using 2푝 timesteps where 푝 = 7, 8, … , 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' In figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 17 to 19 we show accuracy and convergence diagrams for the initial conditions (19) to (21), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' NLS solution for initial condition (21) Figure 20 shows the two different NLS solutions arising from the initial conditions (19) and (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 28 of 33 Stepsizeh10-110 10rERK rERK rERK rEBK4 3 2 1a 10 ErrorERK4 ERK3 ERK2 ERK14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2 1 10 1I t 七11 TT1010210100 Time （se10110210 & 10a 10 Error 6= 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2 1 10 1I t 七11 TT10°11 102TT10°Stepsizeh10-110 10rERK rERK rERK rEBK4 3 2 1a 10 ErrorERK4 ERK3 ERK2 ERK14 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2 1 10 1I t 七11 TT1010210100 Time （se10110210 & 10a 10 Error 6= 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2 1 10 1I t 七11 TT10°11 102TT 10Exponential Runge-Kutta Parareal Error vs Stepsize – Initial Condition (21) Ng = 1 Ng = 2 Ng = 3 Error vs Computational Time – Initial Condition (21) Ng = 1 Ng = 2 Ng = 3 Figure 19: Identical to fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 17 except we are now considering the initial condition (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' NLS Solution – Initial Condition (19) NLS Solution – Initial Condition (21) Figure 20: Solutions of the nonlinear Schrödinger equation (16) for the initial conditions (19) and (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Additional Stability Plots Figures 21 and 22 contain additional stability plots for the Parareal configuration table 3 that show different (푟1, 푟2) ranges than fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Remark regarding convergence region scaling Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Stability regions grow approximately linearly in 푁푔 for small (|푟1|, |푟2|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' To show this, we first assume that we are in a regime where (|푟1|, |푟2|) is small so that the coarse and the fine integrator both exhibit asymptotic error properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' If we construct the coarse propagator \ue233 using 푁푔 steps of a 푞th order integrator, then |푅\ue233| = |||푒푖(푟1+푟2)||| + \ue23b ( |푟1|+|푟2| 푁푔 )푞 < 1 + 퐶1 ( |푟1|+|푟2| 푁푔 )푞 , (62) |푅\ue233 − 푅\ue232| = \ue23b ( |푟1|+|푟2| 푁푔 )푞 < 퐶2 ( |푟1|+|푟2| 푁푔 )푞 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (63) T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 29 of 33 T 714010510X-10010rERK rERK rERK rEBK4 3 2 1a r Error 10ERK4 ERK3 ERK2 ERK14 1 10 t 七10%Stepsize102h10-11010°a 10 Error 61 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='2 1 10 1I t 七11 TT10°100 Time （se11 102101TT 10210 & 10714010510X-100Exponential Runge-Kutta Parareal Parareal Stability Regions and Instability Factors 퐾 = 0 퐾 = 2 퐾 = 4 퐾 = 6 Classical ERK Repartitioned ERK Figure 21: Additional stability regions for the Parareal configuration from table 3 with classical exponential integrators (top column) and repartitioned exponential integrators (bottom column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The gray region is the stability region (35), and color shows the amplification factor |푅(푧1 = 푖푟1, 푧2 = 푖푟2)| outside the stability region where the method is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' These figures show a wider range of 푟2 values than fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The norm of the error matrix can be bounded above by ‖퐄‖∞ = 1 − |푅\ue233|푁푝 1 − |푅\ue233| |푅\ue233 − 푅\ue232| < 푁푝퐶2 ( |푟1|+|푟2| 푁푔 )푞 + \ue23b ( |푟1|+|푟2| 푁푔 )푞 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (64) Convergence is guaranteed if ‖퐄‖∞ < 1, which is equivalent to |푟1| + |푟2| < 1 푁푔퐶2푁푝 + \ue23b ( |푟1|+|푟2| 푁푔 )푞 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (65) Ignoring the higher order terms, the size of the region |푟1| + |푅2| < (퐶2푁푔푁푝)−1 grows linearly in 푁푔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Spatially discretized Vlasov-Poisson equation For notational simplicity we represent the discrete VP solution as the matrix 퐟 where 퐟푗푘 approximates the continuous solution at the grid point (푣푗, 푥푘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Next we define the scaled Fourier wavenumber vectors 퐤푣 = 휋 퐿푣 [0, … , 푁푣∕2 − 1, −푁푣∕2, … , −1], 퐤푥 = 2휋 퐿푥 [0, … , 푁푥∕2 − 1, −푁푥∕2, … , −1], (66) for 퐿푣 = 16, 퐿푥 = 20휋, and the ℝ푁푣,푁푥 matrices 퐊푣 푗푘 = 퐤푣 푗, 퐊푥 푗푘 = 퐤푥 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' If \ue232푣(⋅) and \ue232푥(⋅) represent the discrete Fourier transform in 푣 and 푥, then the transformed variable ̂퐟 = \ue232푥(퐟) satisfies 푑 푑푡 ̂퐟푗푘 = 푣푗푖퐤푥 푘̂퐟푗푘 + \ue232푥 ( 퐄.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' * 휕퐟 휕푣 ) 푗푘 휕퐟 휕푣 = \ue232−1 푥 ( \ue232−1 푣 ( 푖퐊푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' *\ue232푣 ( ̂퐟 ))) (67) T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 30 of 33 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='041015 10100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='041025 1020-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='08 00.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='6Exponential Runge-Kutta Parareal Parareal Stability Regions and Instability Factors 퐾 = 0 퐾 = 2 퐾 = 4 퐾 = 6 Classical ERK Repartitioned ERK Figure 22: Additional stability regions for the Parareal configuration from table 3 with classical exponential integrators (top column) and repartitioned exponential integrators (bottom column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The grey region is the stability region (35), and color shows the amplification factor |푅(푧1 = 푖푟1, 푧2 = 푖푟2)| outside the stability region where the method is unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' These figures show a narrrower range of 푟1 and a wider range of 푟2 than fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' where .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' * denotes the Hadamard product, and the discrete electric field 퐄푗푘 = 퐞푘 ≈ 퐸(푥푘, 푡) is 퐞 = \ue232−1 푥 ( 퐤−퐱.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' * ( 퐛 + Δ푣 푁푣 ∑ 푗=1 ̂퐟푗푘 ) ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ \ue232푥(−1+∫ 20휋 0 푓(푥,푣,푡)푑푣) ) , for 퐛 = \ue232푥(−[1, … , 1]푇 ) ∈ ℝ푁푥, 퐤−푥 푘 = { 0 푗 = 1, 1∕퐤푥 푘 푗 > 1 ∈ ℝ푁푥, Δ푣 = 16∕푁푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' (68) Note that the integral term ∫ 20휋 0 푓(푥, 푣, 푡)푑푣 is treated using the trapezoidal rule, which convergences exponentially on periodic domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' ERK convergence diagrams for KP and Vlassov-Poisson In fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' 23 we show convergence diagrams for the serial ERK methods applied to the KP and VP equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' The plots also contain black crosses that indicate the step-sizes of the coarse and fine integrators for the Parareal configurations described in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 31 of 33 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='041015 10100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='041025 1020-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='08 00.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content='4Exponential Runge-Kutta Parareal Error versus Stepsize – KP equation (53) Ng = 1 Ng = 2 Ng = 3 Fine Integrator Error versus Stepsize – VP Equation (56) Coarse (Ng = 2) Coarse (Ng = 3) Fine Integrator Figure 23: Serial ERK convergence diagrams for the KP and VP equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' We can related these plots to the Parareal configurations described in table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' Specifically, the labeled black crosses at large timesteps correspond to the stepsizes of the coarse integrator, while the black cross on the ERK4 method at the smallest stepsize corresponds to the stepsize of the fine integrator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' T Buvoli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} +page_content=' : Preprint submitted to Elsevier Page 32 of 33 10-10rERK rERK rERK rEBK4 3 2 1Relativ 10 10ERK4 ERK3 ERK2 ERK1TTT TTT 2 Error 10- Stepsize102h1010-810rERK rERK rERK rEBK4 3 2 1Relativ 10ERK4 ERK3 ERK2 ERK1-3Crror e 10- 10Stepsize10°102 hTTT10-' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E2T4oBgHgl3EQfQAYH/content/2301.03764v1.pdf'} diff --git a/8dE3T4oBgHgl3EQfRwni/content/2301.04426v1.pdf b/8dE3T4oBgHgl3EQfRwni/content/2301.04426v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..32cbcf6284e0039da85aa44b0089595257271396 --- /dev/null +++ b/8dE3T4oBgHgl3EQfRwni/content/2301.04426v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid 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a/99FQT4oBgHgl3EQfJzVJ/content/tmp_files/2301.13257v1.pdf.txt b/99FQT4oBgHgl3EQfJzVJ/content/tmp_files/2301.13257v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4501c596f3443419b617beabbac2e0a8839c7607 --- /dev/null +++ b/99FQT4oBgHgl3EQfJzVJ/content/tmp_files/2301.13257v1.pdf.txt @@ -0,0 +1,1689 @@ +arXiv:2301.13257v1 [math.RA] 30 Jan 2023 +CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES +MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP +Abstract. The Fiedler matrices are a large class of companion matrices that include the +well-known Frobenius companion matrix. The Fiedler matrices are part of a larger class +of companion matrices that can be characterized with a Hessenberg form. In this paper, +we demonstrate that the Hessenberg form of the Fiedler companion matrices provides a +straight-forward way to compare the condition numbers of these matrices. We also show +that there are other companion matrices which can provide a much smaller condition num- +ber than any Fiedler companion matrix. We finish by exploring the condition number of a +class of matrices obtained from perturbing a Frobenius companion matrix while preserving +the characteristic polynomial. +1. Introduction +The Frobenius companion matrix is a template that provides a matrix with a prescribed +characteristic polynomial. More recently, it was discovered that the Frobenious companion +matrix belongs to a larger class of Fiedler companion matrices [5], which in turn is a +subset of the intercyclic companion matrices [4]. Other recent templates include nonsparse +companion matrices [2] and generalized companion matrices [6]. +The Frobenius companion matrix is employed in algorithms that use matrix methods to +determine roots of polynomials, but this matrix is not always well-conditioned [3]. Recent +work [3] has explored under what circumstances other Fielder companion matrices can have +a better condition number than the Frobenius matrix, with respect to the Frobenius norm. +After covering background details in Section 2, we use a Hessenberg characterization of the +Fiedler companion matrices in Section 3 to provide a concise argument for the condition +number of a Fielder companion matrix. The characterization allows us to avoid dealing +with the particular permutation in Fiedler’s construction of companion matrices [5], as well +as associated concepts around consecutions and inversions developed in [3]. In Section 4, +we provide some examples of non-Fiedler companion matrices that demonstrate that there +are intercyclic companion matrices that have a smaller condition number than any Fielder +companion matrix for some specific polynomials. In Section 5, we provide a method for con- +structing a generalized companion matrix that, in some cases, can improve on the condition +number of any Fiedler companion matrix. +Date: February 1, 2023. +2010 Mathematics Subject Classification. 15A12, 15B99. +Key words and phrases. companion matrix, Fiedler companion matrix, condition number, generalized +companion matrix. +Research of Vander Meulen was supported in part by NSERC Discovery Grant 2022-05137. +Research of Van Tuyl was supported in part by NSERC Discovery Grant 2019-05412. +Research of Voskamp was supported in part by NSERC USRA 504279. +1 + +2 +MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP + + +0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +−c0 +−c1 +−c2 +−c3 + + , + + +0 +1 +0 +0 +0 +−c3 +1 +0 +0 +−c2 +0 +1 +−c0 +−c1 +0 +0 + + , + + +0 +1 +0 +0 +−c2 +−c3 +1 +0 +0 +0 +0 +1 +−c0 +−c1 +0 +0 + + . +Figure 1. Some 4 × 4 unit sparse companion matrices. + + +0 +1 +0 +0 +−c2 +0 +1 +0 +−c1 + c3c2 +0 +−c3 +1 +−c0 +0 +0 +0 + + , + + +−c3 +1 +0 +0 +0 +0 +1 +0 +−c1 + c3c2 +−c2 +0 +1 +−c0 +0 +0 +0 + + , + + +−c3 +1 +0 +0 +−c2 + a +0 +1 +0 +−c1 + ac3 +−a +0 +1 +−c0 +0 +0 +0 + + . +Figure 2. Some 4 × 4 companion matrices. +2. Technical definitions and background +In this section we recall the relevant background on companion matrices and condition +numbers that will be required throughout the paper. +Let n ≥ 2 be an integer and p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c0. A compan- +ion matrix to p(x) is an n × n matrix A over R[c0, . . . , cn−1] such that the characteristic +polynomial of A is p(x). A unit sparse companion matrix to p(x) is a companion matrix A +that has n − 1 entries equal to one, n variable entries −c0, . . . , −cn−1, and the remaining +n2 − 2n + 1 entries equal to zero. The unit sparse companion matrix of the form + + +0 +1 +0 +· · · +0 +0 +0 +0 +1 +· · · +0 +0 +0 +0 +0 +· · · +1 +0 +... +... +... +· · · +0 +1 +−c0 +−c1 +−c2 +· · · +−cn−2 +−cn−1 + + +is called the Frobenius companion matrix of p(x). Sparse companion matrices have also +been called intercyclic companion matrices due to the structure of the digraph associated +with the matrix (see [7] and [4] for details). +The matrices in Figure 1 are examples of unit sparse companion matrices to p(x) = +x4 + c3x3 + c2x2 + c1x + c0. The first matrix in Figure 1 is a Frobenius companion matrix. +The matrices in Figure 2 are also companion matrices to p(x), but they are not unit sparse +since not every nonzero variable entry is the negative of a single coefficient of p(x). Note +that in the last matrix, the value of a can be any real number; when a = 0, then this matrix +becomes a unit sparse companion matrix. +Since matrix transposition and permutation similarity does not affect the characteristic +polynomial, nor the set of nonzero entries in a matrix, we call two companion matrices +equivalent if one can be obtained from the other via transposition and/or permutation +similarity. +The matrices in Figure 3 are equivalent to the 4 × 4 Frobenius companion +matrix. Note that if A and B are equivalent matrices, then the multiset of entries in any + +CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES +3 + + +−c3 +1 +0 +0 +−c2 +0 +1 +0 +−c1 +0 +0 +1 +−c0 +0 +0 +0 + + , + + +−c3 +−c2 +−c1 +−c0 +1 +0 +0 +0 +0 +1 +0 +0 +0 +0 +1 +0 + + , + + +0 +0 +0 +−c0 +1 +0 +0 +−c1 +0 +1 +0 +−c2 +0 +0 +1 +−c3 + + . +Figure 3. Some companion matrices equivalent to the 4×4 Frobenius com- +panion matrix. +row of A is exactly the multiset of entries of some row or column of B. No two matrices +from Figures 1 and 2 are equivalent (assuming a ̸= 0). +Fielder [5] introduced a class of companion matrices that are constructed as a product +of certain block diagonal matrices. In particular, let F0 be a diagonal matrix with diagonal +entries (1, . . . , 1, −c0) and for k = 1, . . . , n − 1, let +Fk = + + +In−k−1 +O +O +O +Tk +O +O +O +Ik−1 + + with Tk = +� +−ck +1 +1 +0 +� +. +Fiedler showed (see [5, Theorem 2.3]) that the product of these n matrices, in any or- +der, will produce a companion matrix of p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c0. +Consequently, given any permutation σ = (σ0, σ2, . . . , σn−1) of {0, 1, 2, . . . , n − 1}, we say +that Fσ = Fσ0Fσ1 · · · Fσn−1 is a Fiedler companion matrix. The Frobenius companion ma- +trix is a Fiedler companion matrix since the Frobenius companion matrix is equivalent to +F0F1 · · · Fn−1, as noted in [5]. +In [4] it was demonstrated that every unit sparse companion matrix is equivalent to a +unit lower Hessenberg matrix, as summarized in Theorem 2.1. Note that, for 0 ≤ k ≤ n−1, +the k-th subdiagonal of a matrix A = [aij] consists of the entries {ak+1,1, ak+2,2, . . . , an,n−k}. +The 0-th subdiagonal is usually called the main diagonal of a matrix. +Theorem 2.1. [4, Corollary 4.3] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be +a polynomial over R with n ≥ 2. Then A is an n × n unit sparse companion matrix to p(x) +if and only if A is equivalent to a unit lower Hessenberg matrix +(1) +C = + + +0 +Im +O +R +In−m−1 +0T + + +for some (n − m) × (m + 1) matrix R with m(n − 1 − m) zero entries, such that C has +−cn−1−k on its k-th subdiagonal, for 0 ≤ k ≤ n − 1. +Note that in (1), the unit lower Hessenberg matrix C always has Cn,1 = −c0 and R1,m+1 = +−cn−1. Given this Hessenberg characterization of the unit sparse companion matrices, one +can deduce the corresponding inverse matrix if c0 ̸= 0. +Lemma 2.2. [7, Section 7] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be a +polynomial over R with n ≥ 2. +Suppose that C is a unit lower Hessenberg companion + +4 +MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP +matrix to p(x) as in (1). Assuming c0 ̸= 0, if +C = + + +0 +Im +O +u +H +In−m−1 +−c0 +yT +0T + + , for some u, y, H, then C−1 = + + +1 +c0yT +0T +− 1 +c0 +Im +O +0 +− 1 +c0uyT − H +In−m−1 +1 +c0 u + + . +Throughout this paper, we use the Frobenius norm of an n × n matrix A = [ai,j] given +by +||A|| = +�� +i,j +a2 +i,j. +Remark 2.3. If A and B are both unit sparse companion matrices to the same polyno- +mial p(x), then it follows that ||A|| = ||B|| since A and B have exactly the same entries. +Furthermore, if A = PBP T for some permutation matrix P, then A−1 and B−1 also have +the same entries, and hence ||A−1|| = ||B−1||. +The condition number of A, denoted κ(A), is defined to be +κ(A) = ||A|| · ||A−1||. +Remark 2.3 implies the following lemma. +Lemma 2.4. If A and B are equivalent companion matrices, then κ(A) = κ(B). +3. Condition numbers of Fiedler matrices via the Hessenberg +characterization +The condition numbers of Fiedler companion matrices were first calculated by de Ter´an, +Dopico, and P´erez [3, Theorem 4.1]. In this section we demonstrate how a characterization +of Fielder companion matrices via unit lower Hessenberg matrices, as given by Eastman, +et al. [4], provides an efficient way to obtain the condition numbers for Fiedler companion +matrices. +Our approach avoids the use of the consecution-inversion structure sequence, +described in [3, Definition 2.3], which was used in the original computation of these numbers. +The following theorem gives a characterization of the Fielder companion matrices in +terms of unit lower Hesenberg matrices. +Theorem 3.1. [4, Corollary 4.4] If p(x) = xn + cn−1xn−1 + · · · + c1x + c0 is a polynomial +over R with n ≥ 2, then F is an n × n Fiedler companion matrix to p(x) if and only if F is +equivalent to a unit lower Hessenberg matrix as in (1) with the additional property that if +−ck is in position (i, j) then −ck+1 is in position (i − 1, j) or (i, j + 1) for 1 ≤ k ≤ n − 1. +An alternative way to describe the unit lower Hesenberg matrix in Theorem 3.1 is to say +that the variable entries of R in (1) form a lattice-path from the bottom-left corner to the +top-right corner of R. The first two matrices in Figure 1 are examples of Fiedler companion +matrices since the variable entries of R form a lattice-path. The last matrix in Figure 1 is +not a Fiedler companion matrix. +If F is a Fiedler companion matrix, the initial step size of F is the number of coefficients +other than c0 in the row or column containing both c0 and c1. The first matrix in Figure 1 +has initial step size three and the second matrix in Figure 1 has initial step size one. + +CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES +5 +Remark 3.2. Note that equivalent matrices have the same initial step size since transpo- +sitions and permutation equivalence does not change the number of coefficients in the row +or column containing c0 and c1. +Using Theorem 3.1 and Lemma 2.2, one can describe the nonzero entries of the inverse +of a Fiedler companion matrix: +Lemma 3.3. [3, Theorem 3.2] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be +a polynomial over R with n ≥ 2 and c0 ̸= 0. Let F be a Fiedler companion matrix to p(x) +with an initial step size t. Then +(1) F −1 has t + 1 entries equal to − 1 +c0, − c1 +c0, . . . , − ct +c0, +(2) F −1 has n − 1 − t entries equal to ct+1, ct+2, . . . , cn−1, +(3) F −1 has n − 1 entries equal to 1, and +(4) the remaining entries of F −1 are 0. +Proof. Since F is a companion matrix to p(x), by Theorem 2.1, the matrix F is equivalent +to a lower Hessenberg matrix C of the form (1). Since F and C are equivalent, it follows +that the matrices F −1 and C−1 are equivalent, so it suffices to show that the matrix C−1 +satisfies conditions (1) − (4). +Since F is a Fielder companion matrix, Theorem 3.1 implies that c1 is either directly +above c0 in C or directly to the right of c1. If c1 is to right of c0 in C, then all other entries +in the column containing c0 is zero. Alternatively, if c1 is above c0, all entries to the right +of c0 in C are zero. +Lemma 2.2, which gives us the inverse of a unit lower Hessenberg matrix, applies to the +matrix C. By our above observation, the vector u or the vector y must be the zero vector. +Without loss of generality, let yT be zero, which means that − 1 +c0uyT − H = −H. If the +initial step size of A is t, then there will be t nonzero elements in u, and it will have the +form +u = + + +0 +... +0 +−ct +... +−c1 + + +. +By Lemma 2.2 the inverse of the matrix C then has the form +(2) +C−1 = + + +0T +0T +− 1 +c0 +Im +O +0 +−H +In−m−1 +1 +c0u + + +. +From (2), we can describe the entries of C−1: m + n − m − 1 = n − 1 entries are 1 (coming +from the submatrices Im and In−m−1); ct+1, . . . , cn−1, which all belong to the submatrix +−H; the entry − 1 +c0 from the top-right corner; and the entries − c1 +c0 , . . . , − ct +c0 from the term + +6 +MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP +1 +c0u. Moreover, the rest of the entries of C−1 are zero. We have now shown that C−1, and +hence F −1, has the desired properties. +□ +Remark 3.4. Lemma 3.3 mimics [3, Theorem 3.2]. As observed in [7], the initial step size +of a Fiedler companion matrix is equal to the number of initial consecutions or inversions +of the permuation associated with the Fielder companion matrix, as defined in [3]. +We can now compute the condition number for any Fiedler companion matrix. This +result first appeared in [3], but we can avoid the formal analysis of the permutation that +was used to construct the Fiedler companion matrix, as well as the associated concepts of +consecution and inversion of a permutation. +Theorem 3.5. [3, Theorem 4.1] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be +a polynomial over R with n ≥ 2 and c0 ̸= 0. Let F be a Fiedler companion matrix to p(x) +with an initial step size t. Then +κ(F)2 = ||F||2 · +� +(n − 1) + 1 + |c1|2 + · · · + |ct|2 +|c0|2 ++ |ct+1|2 + · · · + |cn−1|2 +� +, +with +||F||2 = (n − 1) + |c0|2 + |c1|2 + · · · + |cn−1|2. +Proof. This result follows from the fact that F is a unit sparse companion matrix (so it +contains n − 1 entries equal to 1 and the entries −c0, . . . , −cn−1), and Lemma 3.3, which +describes the entries of F −1. +□ +Because the condition number κ(F) of a Fiedler companion matrix F depends only upon +the initial step size and not the permutation σ, we can derive the following corollary. +Corollary 3.6. [3, Corollary 4.3] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be +a polynomial over R with n ≥ 2 and c0 ̸= 0. Let A and B be Fiedler companion matrices +to the polynomial p(x). If the initial step size of both A and B is t, then κ(A) = κ(B). +Since condition numbers of Fiedler companion matrices depend on the initial step size, +let +St = {F | F is a Fiedler companion matrix to p(x) with initial step size t}, +and define κ(t) = κ(F) for F ∈ St. We can now recover a result of [3] that allows us to +compare the condition numbers of Fielder matrices while again avoiding any reference to +the permutation σ used to define a Fiedler matrix. +Corollary 3.7. [3, Corollary 4.5] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be +a polynomial over R with n ≥ 2 and c0 ̸= 0. Then +(1) if |c0| < 1, then κ(1) ≤ κ(2) ≤ · · · ≤ κ(n − 1); +(2) if |c0| = 1, then κ(1) = κ(2) = · · · = κ(n − 1); and +(3) if |c0| > 1, then κ(1) ≥ κ(2) ≥ · · · ≥ κ(n − 1). +Proof. Note that by Corollary 3.6, κ(A) is the same for all A ∈ St, so κ(t) is well-defined. +The conclusions follow from Theorem 3.5. +□ + +CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES +7 +One of our new results is to compare the condition number of a Fielder companion +matrix of p(x) to the condition number of other companion matrices of p(x). In particular, +if a Fiedler companion matrix F has a smaller condition number than another companion +matrix C to the same polynomial p(x), then the ratio κ(C) +κ(F ) can be bounded. This result is +similar in spirit to [3, Theorem 4.12]. +Theorem 3.8. Let p(x) = xn + cn−1xn−1 + · · · + c1x + c0 be a polynomial over R with +n ≥ 2, and c0 ̸= 0. Let F be a Fielder companion matrix to p(x). Further, suppose C is +any companion matrix to p(x) whose lower Hessenberg form is +C = + + +0 +Im +O +uC +HC +In−m−1 +−c0 +yT +C +0T + + +such that either uC or yT +C is the zero vector. If κ(F) ≤ κ(C), then +1 ≤ κ(C) +κ(F) ≤ κ(F). +Proof. The conclusion that 1 ≤ κ(C) +κ(F ) is immediate from the hypothesis that κ(F) ≤ κ(C). +By Theorem 3.1 and Lemma 2.4, we can assume F is in unit lower Hessenberg form. As +such, let +F = + + +0 +Il +O +uF +HF +In−l−1 +−c0 +yT +F +0T + + +. +and let t be the initial step size of F. We want to show that +||C|| · ||C−1|| +||F|| · ||F −1|| ≤ ||F|| · ||F −1||. +Since C and F are unit sparse companion matrices, ||C|| = ||F||. It suffices to show that +||C−1|| ≤ ||F|| · ||F −1||2. +Using equivalence, we may assume without loss of generality that uC = 0. By Lemma 2.2, +C−1 = + + +1 +c0yT +C +0T +− 1 +c0 +Im +O +0 +−HC +In−m−1 +0 + + +. +since uC = 0. Then +(3) +||C−1||2 = (n − 1) + +� 1 +c0 +�2 ++ +� +ci∈yT +C +���� +ci +c0 +���� +2 ++ +� +ck∈HC +|ck|2. + +8 +MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP +where c ∈ H (resp. c ∈ y) means −c is an entry in H (resp. y). On the other hand, using +Lemma 3.3, +(4) ||F||2 · ||F −1||4 = +� +(n − 1) + +n−1 +� +i=0 +|ci|2 +�  +(n − 1) + +� 1 +c0 +�2 ++ +t +� +i=1 +���� +ci +c0 +���� +2 ++ +n−1 +� +j=t+1 +|cj|2 + + +2 +. +We want to show that ||C−1|| ≤ ||F||·||F −1||2 which is equivalent to showing that ||C−1||2 ≤ +||F||2 · ||F −1||4. To do this, for each of the four different summands in (3), we show that +there exists distinct terms in ||F||2 ·||F −1||4 that are greater than or equal to the summand. +Here we rely on the fact that there are no negative summands in (4). +Partially expanding out (4), we have +||F||2 · ||F −1||4 = (n − 1)3 + (n − 1)2 +� 1 +c0 +�2 ++ (n − 1) +�n−1 +� +i=0 +|ci|2 +� � 1 +c0 +�2 ++ (n − 1)2 +n−1 +� +j=0 +|cj|2 + other non-negative terms. +Consequently, +||C−1||2 += +(n − 1) + +� 1 +c0 +�2 ++ +� +ci∈yT +C +���� +ci +c0 +���� +2 ++ +� +ck∈HC +|ck|2 +≤ +(n − 1)3 + (n − 1)2 +� 1 +c0 +�2 ++ (n − 1) +�n−1 +� +i=0 +|ci|2 +� � 1 +c0 +�2 ++ (n − 1)2 +n−1 +� +j=0 +|cj|2 +≤ +||F||2 · ||F −1||4. +□ +4. Striped Companion Matrices +In this section we explore a particular class of companion matrices known as striped +companion matrices, which were introduced in [4]. +A striped companion matrix to a +polynomial p(x) = xn + cn−1xn−1 + · · · + c1x + c0 has the property that the coefficients +−c0, −c1, . . . , −cn−1 form horizontal stripes in the matrix. In particular, if t = (t1, t2, . . . , tr) +is an ordered r-tuple of positive integers with t1 +t2 +· · ·+tr = n, and t1 ≥ ti for 2 ≤ i ≤ n, +then we define the striped companion matrix Sn(t) to be the companion matrix of unit Hes- +senberg form +Sn(t) = + + +0 +It1−1 +O +R +In−t1 +0T + + +(5) +with the (n − t1 + 1) × t1 matrix R having r nonzero rows and with the ith nonzero row of +R having ti variables in the first ti positions and ti − 1 zero rows immediately above it in + +CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES +9 +R, for 1 < i ≤ r. Note that this implies the first row of R is a nonzero row with t1 leading +nonzero entries. For example, +S7(3, 2, 2) = +� +���������� +0 +1 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +−c4 +−c5 +−c6 +1 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +−c2 +−c3 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +1 +−c0 +−c1 +0 +0 +0 +0 +0 +� +���������� +, and S8(3, 3, 2) = +� +������������ +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +−c5 +−c6 +−c7 +1 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +−c2 +−c3 +−c4 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +1 +−c0 +−c1 +0 +0 +0 +0 +0 +0 +� +������������ +. +As the next theorem shows, in some cases the stripped companion matrices can have a +better condition number than a Fielder companion matrix. +Theorem 4.1. Suppose n = k(m + 1) for some positive k, m ∈ Z and p(x) = xn + +cn−1xn−1 + · · · + c1x + c0 with c0 = 1, c1, . . . , cn−1 ∈ R. There exists a striped companion +matrix S = Sn(k, . . . , k) for p(x) such that κ(S) ≤ κ(F) for every Fiedler companion matrix +F if and only if +(6) +m +� +j=1 +�k−1 +� +i=1 +|cicjk − cjk+i|2 +� +≤ +m +� +j=1 +�k−1 +� +i=1 +|cjk+i|2 +� +. +Proof. Let S = Sk(m+1)(k, . . . , k), and let F be a Fiedler companion matrix. Since ||S|| = +||F|| as noted in Remark 2.3, it suffices to show that ||S−1|| ≤ ||F −1|| if and only if equation +(6) holds. By Lemma 2.2, +S−1 = + + +−c1 +−c2 +. . . +−ck−1 +0T +−1 +Ik−1 +O +0 +−c1cmk + cmk+1 +−c2cmk + cmk+2 +. . . +−ck−1cmk + c(m+1)k−1 +0 +0 +. . . +0 +... +... +. . . +... +... +... +. . . +... +0 +0 +. . . +0 +−c1c2k + c2k+1 +−c2c2k + c2k+2 +. . . +−ck−1c2k + c3k−1 +0 +0 +. . . +0 +... +... +. . . +... +0 +0 +. . . +0 +−c1ck + ck+1 +−c2ck + ck+2 +. . . +−ck−1ck + c2k−1 +0 +0 +. . . +0 +... +... +. . . +... +0 +0 +. . . +0 +Imk +−cmk +0 +... +... +0 +−c2k +0 +... +0 +−ck +0 +... +0 + + +. +Thus +||S−1||2 = n + +k−1 +� +j=1 +|cj|2 + +m +� +j=1 +|cjk|2 + +m +� +j=1 +�k−1 +� +i=1 +|cicjk − cjk+i|2 +� +. + +10 +MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP +By Theorem 3.5, +||F −1||2 = n + +k−1 +� +j=1 +|cj|2 + +m +� +j=1 +|cjk|2 + +m +� +j=1 +�k−1 +� +i=1 +|cjk+i|2 +� +. +Therefore κ(S) ≤ κ(F) if and only if +m +� +j=1 +�k−1 +� +i=1 +|cicjk − cjk+i|2 +� +≤ +m +� +j=1 +�k−1 +� +i=1 +|cjk+i|2 +� +. +□ +We can deduce the following corollary. +Corollary 4.2. Suppose n = k(m + 1) for some m, k ∈ Z and p(x) = xn + cn−1xn−1 + +· · · + c1x + c0 with c0 = 1, c1, . . . , cn−1 ∈ R. Suppose F is any Fiedler companion matrix +for p(x). If +|cicjk − cjk+i| ≤ |cjk+i|, for 1 ≤ j ≤ m and 1 ≤ i ≤ k − 1, +then there exists a striped companion matrix S = Sn(k, . . . , k), such that κ(S) ≤ κ(F). +Example 4.3. Let +p(x) = x9 + 8x8 + 6x7 + 2x6 + 5x5 + 8x4 + 3x3 + 3x2 + 2x + 1. +Note that the inequalities in Corollary 4.2 hold. Let F be any Fiedler companion to p(x) +and consider the striped companion matrix S = S9(3, 3, 3), i.e., +S9(3, 3, 3) = + + +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +−2 +−6 +−8 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +−3 +−8 +−5 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +0 +0 +0 +0 +0 +0 +0 +1 +−1 +−2 +−3 +0 +0 +0 +0 +0 +0 + + +. +Then ||S|| = ||F|| = +√ +224, but κ(S) = +√ +224 +√ +63 < κ(F) = +√ +224 +√ +224. +One extreme example of how the inequalities in Corollary 4.2 can be met is if c0 = 1 and +the striped companion matrix in line (5) has rank(R) = 1. In this case, the inequalities are +trivially met as described in the following corollary. A more general result can be developed +for striped companion matrices with differing stripe lengths; e.g., see [1, Section 4.2]. +Corollary 4.4. Given p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 with c0 = 1, and +c1, . . . , cn−1 ∈ R, let S be a striped companion matrix to the polynomial p(x). If +S = + + +0 Im +O +R +In−m−1 +0T + + +with rank(R) = 1, then κ(S) ≤ κ(F) for any Fiedler companion matrix F. + +CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES +11 +Proof. This result follows from Corollary 4.2 by observing that |cicjk − cjk+i| = 0 for all +1 ≤ j ≤ m and 1 ≤ i ≤ k − 1, if and only if rank(R) = 1. In particular, rank(R) = 1 +if and only every 2 × 2 submatrix of R has determinant zero, which is true if and only if +|cicjk − cjk+i| = 0 for 1 ≤ j ≤ m and 1 ≤ i ≤ k − 1. Note that we are using the fact that +� +−cjk +−cjk+i +−c0 +−ci +� +is a 2 × 2 submatrix of R and c0 = 1. +□ +Example 4.5. Let b, k ∈ R and consider the polynomial p(x) = x6 + (bk3)x5 + (bk2)x4 + +(bk2)x3 + (bk)x2 + kx + 1. If +S = S6(2, 2, 2) = + + +0 +1 +0 +0 +0 +0 +−bk2 +−bk3 +1 +0 +0 +0 +0 +0 +0 +1 +0 +0 +−bk +−bk2 +0 +0 +1 +0 +0 +0 +0 +0 +0 +1 +−1 +−k +0 +0 +0 +0 + + +and F is any Fiedler companion matrix for p(x), then +�κ(F) +κ(S) +�2 += b2k6 + b2k4 + b2k4 + b2k2 + k2 + 6 +b2k4 + b2k2 + k2 + 6 +. +In this case, for sufficiently large k, +κ(F) +κ(S) ≈ k +demonstrating a significantly better condition number for S compared to any Fiedler com- +panion matrix. +As shown in Corollary 4.4, if the rank of the submatrix R in the striped companion matrix +S has rank(R) = 1, then the inequality κ(S) ≤ κ(F) holds for any Fiedler companion matrix +F. Note that in the striped companion matrix given in Example 4.5, the corresponding +submatrix R has rank one. Observe also that we can write p(x) has +p(x) = q(x) + (bk)x2q(x) + (bk2)x4q(x) + x6 with q(x) = 1 + kx. +This generalizes: if the matrix S in Corollary 4.4 has rank(R) = 1, then p(x) = xn+q(x)f(x) +for some polynomial q(x) with deg(q(x)) = m and deg(f(x)) = n − m − 1. Moreover, +Corollary 4.4 can be improved by giving an estimate on κ(F ) +κ(S) for any Fiedler companion +matrix F. +Theorem 4.6. Suppose n = k(m + 1) and p(x) = q(x) + b1xkq(x) + b2x2kq(x) + · · · + +bmxmkq(x) + x(m+1)k with +q(x) = ak−1xk−1 + ak−2xk−2 + · · · + a1x + 1. +Let S = Sn(k, k, . . . , k) and F be any Fiedler companion matrix to p(x). If (b2 +1 + · · · + b2 +m) +is sufficiently large, then +�κ(F) +κ(S) +�2 +≈ (a2 +1 + · · · + a2 +k−1 + 1), + +12 +MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP +or if (a2 +1 + · · · + a2 +k−1) is sufficiently large, then +�κ(F) +κ(S) +�2 +≈ (1 + b2 +1 + · · · + b2 +m). +Proof. By Remark 2.3, κ(F ) +κ(S) = ||F −1|| +||S−1||. By Lemma 2.2, +||S−1||2 = a2 +1 + · · · + a2 +k−1 + b2 +1 + · · · + b2 +m + n. +By Theorem 3.5 we can determine that +||F −1||2 = (1 + b2 +1 + · · · + b2 +m)(a2 +1 + · · · + a2 +k−1) + (b2 +1 + · · · + b2 +m) + n. +Therefore, +�κ(F) +κ(S) +�2 += (1 + b2 +1 + · · · + b2 +m)(a2 +1 + · · · + a2 +k−1) + (b2 +1 + · · · + b2 +m) + n +(a2 +1 + · · · + a2 +k−1) + (b2 +1 + · · · + b2m) + n +and the result follows. +□ +5. Generalized companion matrices: a case study +In the previous sections, we focused on the condition numbers of unit sparse companion +matrices. In this section, we initiate an investigation into the condition numbers of a family +of matrices that are not companion matrices, but have properties similar to companion +matrices. To date, there appears to be little work done on this approach, so the work in +this section can be seen as providing a proof-of-concept for future projects. These results can +also be viewed in the broader context of developing the properties of generalized companion +matrices (e.g., see [4, 6]). Roughly speaking, given a polynomial p(x) = xn + cn−1xn−1 + +· · ·+c1x1 +c0, a generalized companion matrix A is a matrix whose entries are polynomials +in the c0, . . . , cn and whose characteristic polynomial is p(x). See [6] for more explicit detail. +Instead of considering the general case, we focus on a particular family of matrices and +their condition numbers. This case study shows that the condition numbers can improve +on those of Frobenius (or Fiedler) companion matrices under some extra hypotheses. +We now define our special family. Let p(x) = xn+cn−1xn−1+· · ·+c1x+c0 be a polynomial +over R with n ≥ 2 and let a ∈ R be any real number. Fix an integer ℓ ∈ {3, . . . , n − 2} and +let +aT += +(−cn−1, −cn−2, . . . , −cℓ+1) and bT = (−cℓ−2, −cℓ−3, . . . , −c1). +Then let +(7) +Mn(a, ℓ) = + + +a +In−ℓ−1 +O +O +−cℓ + a +W +I2 +O +−cℓ−1 + acn−1 +b +O +O +Iℓ−2 +−c0 +O +O +O + + +. + +CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES +13 + + +−c6 +1 +0 +0 +0 +0 +0 +−c5 +0 +1 +0 +0 +0 +0 +−c4 + a +0 +0 +1 +0 +0 +0 +−c3 + ac6 +−a +0 +0 +1 +0 +0 +−c2 +0 +0 +0 +0 +1 +0 +−c1 +0 +0 +0 +0 +0 +1 +−c0 +0 +0 +0 +0 +0 +0 + + +Figure 4. The matrix M7(a, 4) +where W is a 2 × (n − ℓ − 1) matrix having W2,1 = −a and zeroes in every other entry. +Informally, the matrix Mn(a, ℓ) is constructed by starting with the Frobenius companion +matrix which has all the coefficents of p(x) in the first column. Then we fix a row that is +neither the top row nor one of the bottom two rows (this corresponds to picking the ℓ), and +then adding a to cℓ in the (n − ℓ)-th row, and −a in the column to the right and one below. +We then also add acn−1 to the first entry in the (n − ℓ + 1)-th row. Note that when a = 0, +Mn(0, ℓ) is equivalent to the Frobenius companion matrix. We can thus view Mn(a, ℓ) as a +perturbation of the Frobenius companion matrix when a ̸= 0. As an example, the matrix +M7(a, 4) is given in Figure 4. +We wish to compare the condition number of Mn(a, ℓ) with the Frobenius (and Fiedler) +companion matrices. In some cases our new matrix Mn(a, ℓ) can provide us with a smaller +condition number. The next lemma gives the inverse of Mn(a, ℓ) and shows that the char- +acteristic polynomial of Mn(a, ℓ) is p(x). +Lemma 5.1. Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be a polynomial over +R, with n ≥ 2 and c0 ̸= 0. Let a ∈ R and ℓ ∈ {3, . . . , n − 2}, and let M = Mn(a, ℓ) be +constructed from p(x) as above. Then +(i) the characteristic polynomial of M is p(x), and +(ii) if c0 ̸= 0, then +M−1 = 1 +c0 + + +0T +0T +0T +−1 +c0In−ℓ +O +O +a +−c0W +c0I2 +O +−cℓ + a +−cℓ−1 +O +O +c0Iℓ−2 +b + + +. +Proof. (i) We employ the fact that the determinant of a matrix is a linear function of its +rows. In particular, if M = Mn(a, ℓ), we observe that row n − ℓ of xIn − M can be written + +14 +MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP +as u + av for some vectors u and v such that u is not a function of a. Row n − ℓ + 1 of +xIn − M can also be written in a similar manner. Let k = n − ℓ. Thus applying linearity to +row k gives us +det(xIn − M) = det + + + + + + + +xIn − + + +a +In−ℓ−1 +O +O +−cℓ +W +I2 +O +−cℓ−1 + acn−1 +b +O +O +Iℓ−2 +−c0 +O +O +O + + + + + + + + + ++ a(−1)xℓ. +(8) +Note that the term a(−1)xℓ in (8) comes from computing the determinant of the matrix A′ +formed by replacing the k-th row of the matrix xIn−M with the row +� +−a +0 +· · · +0 +� +. Do- +ing a row expansion along the k-th row of A′, the determinant of A′ is (−1)k+1(−a)det(A′′) +where A′′ is a block lower diagonal matrix with diagonal blocks D1 and D2. Furthermore, +D1 is a (k−1)×(k−1) lower triangular matrix with −1 on all the diagonal entries, and D2 is +a ℓ × ℓ upper triangular matrix with x on all the diagonal entries. So det(A′′) = (−1)k−1xℓ, +and hence det(A′) = (−1)k+1(−a)(−1)k−1xℓ = (−a)xℓ, as desired. +We now apply linearity to row k + 1 in the matrix that appears on the right-hand side +of (8); in particular, a similar argument shows that the right-hand side (8) is equal to +(9) +det + + + + + + + +xI − + + +a +Ik−1 +O +O +−cℓ +O +I2 +O +−cℓ−1 +b +O +O +Iℓ−2 +−c0 +O +O +O + + + + + + + + + ++a(−1)xℓ +acn−1(−1)xℓ−1 +a(x+cn−1)xℓ−1. +Note that the first summand in (9) is the characteristic polynomial of a Frobenius companion +matrix of p(x), and hence is p(x). Thus, (9) reduces to +p(x) + a(−1)xℓ + acn−1(−1)xℓ−1 + a(x + cn−1)xℓ−1 = p(x). +(ii) A direct multiplication will show that the given matrix is the inverse M. +□ +Because both Mn(a, ℓ) and its inverse are known, we are able to compute its condition +number. In the next lemma, instead of providing the general formula, we compute the +condition number under the extra assumption that c0 = 1 in the polynomial p(x). +Lemma 5.2. Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be a polynomial over +R, with n ≥ 2, and suppose that c0 = 1. +Let a ∈ R and ℓ ∈ {3, . . . , n − 2}, and let +M = Mn(a, ℓ). Then +κ(M)2 = +� +v + a2 + (a − cℓ)2 + (acn−1 − cℓ−1)2� � +v + a2 + (a − cℓ)2 + c2 +ℓ−1 + 1 +� +with +v = n − c2 +ℓ−1 − c2 +ℓ + +n−1 +� +i=1 +c2 +i . + +CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES +15 +The next result illustrates the desired proof-of-concept. In particular, the result shows +that in special cases, the condition number of the matrix Mn(a, ℓ), which has properties +similar to a companion matrix, has a condition number smaller than any Fielder compan- +ion matrix. Although the scope of this result is limited, it does suggest that generalized +companion matrices, and in particular perturbations of the Frobenius companion matrix, +can provide better condition numbers in some cases. +Theorem 5.3. Let n ≥ 2, and fix ℓ ∈ {3, . . . , n − 2} and t ∈ R. Set +p(x) = xn + txn−1 + txℓ + t2xℓ−1 + 1. +Let M = Mn(t, ℓ). Then, for any Fieldler companion matrix F of p(x), +κ(F)2 +κ(M)2 = +(n + 2t2 + t4)2 +(n + 2t2)(n + 1 + 2t2 + t4). +In particular, for t for sufficiently large, +κ(F ) +κ(M) ≈ +1 +√ +2t. +Proof. By Lemma 5.2, +κ(M)2 = +� +1 + t2 + (n − 1) + a2 + (a − t)2 + (at − t2)2� � +1 + t2 + (n − 1) + a2 + (a − t)2 + t4 + 1 +� +. +Setting a = t gives κ(M)2 = (n + 2t2)(n + 1 + 2t2 + t4). We use Theorem 3.5 to compute +κ(F)2. Note that since c0 = 1, κ(F) is independent of the initial step size of F. Hence +κ(F) += +((n − 1) + 1 + t4 + t2 + t2) = (n + 2t2 + t4). +Thus we have +κ(F)2 +κ(M)2 = +(n + 2t2 + t4)2 +(n + 2t2)(n + 1 + 2t2 + t4). +The limit of the right hand side is t2 +2 as t → ∞, which implies the final statement. +□ +The following result gives another case where we can make a matrix with smaller condi- +tion number than any other Fielder companion matrix, providing additional evidence that +generalized companion matrices may be of interest. +Theorem 5.4. Let n ≥ 2, and fix ℓ ∈ {3, . . . , n−2}. Let p(x) = xn+cn−1xn−1+· · ·+c1x+c0 +with c0 = 1, and (cℓcn−1)2 < 2cℓ−1cℓcn−1 − 1. Let M = Mn(cℓ, ℓ). Then κ(M) < κ(F) for +every Fieldler companion matrix F of p(x). +Proof. Let v = n − c2 +ℓ − c2 +ℓ−1 + �n−1 +i=1 c2 +i . +Because c0 = 1, by Theorem 3.5 all Fielder +companion matrices F have condition number +κ(F) = (v + c2 +ℓ + c2 +ℓ−1). +By Lemma 5.2, with a = cℓ, +κ(M)2 += +(v + c2 +ℓ + (cℓcn−1 − cℓ−1)2)(v + c2 +ℓ + c2 +ℓ−1 + 1) += +(v + c2 +ℓ + c2 +ℓ−1 + ((cℓcn−1)2 − 2cℓ−1cℓcn−1))(v + c2 +ℓ + c2 +ℓ−1 + 1). +If we set w = (v +c2 +ℓ +c2 +ℓ−1), then κ(M)2 = (w−y)(w+1) with y = 2cℓ−1cℓcn−1 −(cℓcn−1)2. +But y > 1 by hypothesis, thus κ(M)2 < w2 = κ(F)2. +□ + +16 +MICHAEL COX, KEVIN N. VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP +References +[1] M. Cox, On conditions numbers of companion matrices, M.Sc. Thesis, McMaster University, 2018. +[2] L. Deaett, J. Fischer, C. Garnett, K.N. Vander Meulen, Non-sparse companion matrices, Electron. J. +Linear Algebra 35 (2019) 223–247. +[3] F. de Ter´an, F.M. Dopico, J. P´erez, Condition numbers for inversion of Fiedler companion matrices, +Linear Algebra Appl. 439 (2013) 944–981. +[4] B. Eastman, I.J. Kim, B.L. Shader, K.N. Vander Meulen, Companion matrix patterns, Linear Algebra +Appl. 463 (2014) 255–272. +[5] M. Fiedler, A note on companion matrices, Linear Algebra Appl. 372 (2003) 325–331. +[6] C. Garnett, B.L. Shader, C.L. Shader, P. van den Driessche, Characterization of a family of generalized +companion matrices, Linear Algebra Appl. 498 (2016) 360–365. +[7] K.N. Vander Meulen, T. Vanderwoerd, Bounds on polynomial roots using intercyclic companion matri- +ces, Linear Algebra Appl. 539 (2018) 94–116. +Unit 202 - 133 Herkimer Street, Hamilton, ON, L8P 2H3, Canada +Email address: michael.m.cox@outlook.com +Department of Mathematics, Redeemer University College, Ancaster, ON, L9K 1J4, Canada +Email address: kvanderm@redeemer.ca +Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4L8, +Canada +Email address: vantuyl@math.mcmaster.ca +Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4L8, +Canada +Email address: voskampj@mcmaster.ca + diff --git a/99FQT4oBgHgl3EQfJzVJ/content/tmp_files/load_file.txt b/99FQT4oBgHgl3EQfJzVJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..70c3c97a674cc46501298fa0003495223bd26577 --- /dev/null +++ b/99FQT4oBgHgl3EQfJzVJ/content/tmp_files/load_file.txt @@ -0,0 +1,611 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf,len=610 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='13257v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='RA] 30 Jan 2023 CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES MICHAEL COX, KEVIN N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The Fiedler matrices are a large class of companion matrices that include the well-known Frobenius companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The Fiedler matrices are part of a larger class of companion matrices that can be characterized with a Hessenberg form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In this paper, we demonstrate that the Hessenberg form of the Fiedler companion matrices provides a straight-forward way to compare the condition numbers of these matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' We also show that there are other companion matrices which can provide a much smaller condition num- ber than any Fiedler companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' We finish by exploring the condition number of a class of matrices obtained from perturbing a Frobenius companion matrix while preserving the characteristic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Introduction The Frobenius companion matrix is a template that provides a matrix with a prescribed characteristic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' More recently, it was discovered that the Frobenious companion matrix belongs to a larger class of Fiedler companion matrices [5], which in turn is a subset of the intercyclic companion matrices [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Other recent templates include nonsparse companion matrices [2] and generalized companion matrices [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The Frobenius companion matrix is employed in algorithms that use matrix methods to determine roots of polynomials, but this matrix is not always well-conditioned [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Recent work [3] has explored under what circumstances other Fielder companion matrices can have a better condition number than the Frobenius matrix, with respect to the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' After covering background details in Section 2, we use a Hessenberg characterization of the Fiedler companion matrices in Section 3 to provide a concise argument for the condition number of a Fielder companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The characterization allows us to avoid dealing with the particular permutation in Fiedler’s construction of companion matrices [5], as well as associated concepts around consecutions and inversions developed in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In Section 4, we provide some examples of non-Fiedler companion matrices that demonstrate that there are intercyclic companion matrices that have a smaller condition number than any Fielder companion matrix for some specific polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In Section 5, we provide a method for con- structing a generalized companion matrix that, in some cases, can improve on the condition number of any Fiedler companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Date: February 1, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 15A12, 15B99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' companion matrix, Fiedler companion matrix, condition number, generalized companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Research of Vander Meulen was supported in part by NSERC Discovery Grant 2022-05137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Research of Van Tuyl was supported in part by NSERC Discovery Grant 2019-05412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Research of Voskamp was supported in part by NSERC USRA 504279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 1 2 MICHAEL COX, KEVIN N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 1 0 0 0 0 1 −c0 −c1 −c2 −c3 \uf8f9 \uf8fa\uf8fa\uf8fb , \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 0 −c3 1 0 0 −c2 0 1 −c0 −c1 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 −c2 −c3 1 0 0 0 0 1 −c0 −c1 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Some 4 × 4 unit sparse companion matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' \uf8ee \uf8ef\uf8ef\uf8f0 0 1 0 0 −c2 0 1 0 −c1 + c3c2 0 −c3 1 −c0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , \uf8ee \uf8ef\uf8ef\uf8f0 −c3 1 0 0 0 0 1 0 −c1 + c3c2 −c2 0 1 −c0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , \uf8ee \uf8ef\uf8ef\uf8f0 −c3 1 0 0 −c2 + a 0 1 0 −c1 + ac3 −a 0 1 −c0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Some 4 × 4 companion matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Technical definitions and background In this section we recall the relevant background on companion matrices and condition numbers that will be required throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let n ≥ 2 be an integer and p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' A compan- ion matrix to p(x) is an n × n matrix A over R[c0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , cn−1] such that the characteristic polynomial of A is p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' A unit sparse companion matrix to p(x) is a companion matrix A that has n − 1 entries equal to one, n variable entries −c0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , −cn−1, and the remaining n2 − 2n + 1 entries equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The unit sparse companion matrix of the form \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 1 0 · · 0 0 0 0 1 · · 0 0 0 0 0 · · 1 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' · · 0 1 −c0 −c1 −c2 · · −cn−2 −cn−1 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb is called the Frobenius companion matrix of p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Sparse companion matrices have also been called intercyclic companion matrices due to the structure of the digraph associated with the matrix (see [7] and [4] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The matrices in Figure 1 are examples of unit sparse companion matrices to p(x) = x4 + c3x3 + c2x2 + c1x + c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The first matrix in Figure 1 is a Frobenius companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The matrices in Figure 2 are also companion matrices to p(x), but they are not unit sparse since not every nonzero variable entry is the negative of a single coefficient of p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Note that in the last matrix, the value of a can be any real number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' when a = 0, then this matrix becomes a unit sparse companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Since matrix transposition and permutation similarity does not affect the characteristic polynomial, nor the set of nonzero entries in a matrix, we call two companion matrices equivalent if one can be obtained from the other via transposition and/or permutation similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The matrices in Figure 3 are equivalent to the 4 × 4 Frobenius companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Note that if A and B are equivalent matrices, then the multiset of entries in any CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES 3 \uf8ee \uf8ef\uf8ef\uf8f0 −c3 1 0 0 −c2 0 1 0 −c1 0 0 1 −c0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fb , \uf8ee \uf8ef\uf8ef\uf8f0 −c3 −c2 −c1 −c0 1 0 0 0 0 1 0 0 0 0 1 0 \uf8f9 \uf8fa\uf8fa\uf8fb , \uf8ee \uf8ef\uf8ef\uf8f0 0 0 0 −c0 1 0 0 −c1 0 1 0 −c2 0 0 1 −c3 \uf8f9 \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Some companion matrices equivalent to the 4×4 Frobenius com- panion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' row of A is exactly the multiset of entries of some row or column of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' No two matrices from Figures 1 and 2 are equivalent (assuming a ̸= 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Fielder [5] introduced a class of companion matrices that are constructed as a product of certain block diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In particular, let F0 be a diagonal matrix with diagonal entries (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , 1, −c0) and for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , n − 1, let Fk = \uf8ee \uf8f0 In−k−1 O O O Tk O O O Ik−1 \uf8f9 \uf8fb with Tk = � −ck 1 1 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Fiedler showed (see [5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='3]) that the product of these n matrices, in any or- der, will produce a companion matrix of p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Consequently, given any permutation σ = (σ0, σ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , σn−1) of {0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , n − 1}, we say that Fσ = Fσ0Fσ1 · · · Fσn−1 is a Fiedler companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The Frobenius companion ma- trix is a Fiedler companion matrix since the Frobenius companion matrix is equivalent to F0F1 · · · Fn−1, as noted in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In [4] it was demonstrated that every unit sparse companion matrix is equivalent to a unit lower Hessenberg matrix, as summarized in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Note that, for 0 ≤ k ≤ n−1, the k-th subdiagonal of a matrix A = [aij] consists of the entries {ak+1,1, ak+2,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , an,n−k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The 0-th subdiagonal is usually called the main diagonal of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' [4, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='3] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be a polynomial over R with n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Then A is an n × n unit sparse companion matrix to p(x) if and only if A is equivalent to a unit lower Hessenberg matrix (1) C = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 Im O R In−m−1 0T \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb for some (n − m) × (m + 1) matrix R with m(n − 1 − m) zero entries, such that C has −cn−1−k on its k-th subdiagonal, for 0 ≤ k ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Note that in (1), the unit lower Hessenberg matrix C always has Cn,1 = −c0 and R1,m+1 = −cn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Given this Hessenberg characterization of the unit sparse companion matrices, one can deduce the corresponding inverse matrix if c0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' [7, Section 7] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be a polynomial over R with n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Suppose that C is a unit lower Hessenberg companion 4 MICHAEL COX, KEVIN N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP matrix to p(x) as in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Assuming c0 ̸= 0, if C = \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 0 Im O u H In−m−1 −c0 yT 0T \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb , for some u, y, H, then C−1 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 1 c0yT 0T − 1 c0 Im O 0 − 1 c0uyT − H In−m−1 1 c0 u \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Throughout this paper, we use the Frobenius norm of an n × n matrix A = [ai,j] given by ||A|| = �� i,j a2 i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' If A and B are both unit sparse companion matrices to the same polyno- mial p(x), then it follows that ||A|| = ||B|| since A and B have exactly the same entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Furthermore, if A = PBP T for some permutation matrix P, then A−1 and B−1 also have the same entries, and hence ||A−1|| = ||B−1||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The condition number of A, denoted κ(A), is defined to be κ(A) = ||A|| · ||A−1||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='3 implies the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' If A and B are equivalent companion matrices, then κ(A) = κ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Condition numbers of Fiedler matrices via the Hessenberg characterization The condition numbers of Fiedler companion matrices were first calculated by de Ter´an, Dopico, and P´erez [3, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In this section we demonstrate how a characterization of Fielder companion matrices via unit lower Hessenberg matrices, as given by Eastman, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' [4], provides an efficient way to obtain the condition numbers for Fiedler companion matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Our approach avoids the use of the consecution-inversion structure sequence, described in [3, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='3], which was used in the original computation of these numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The following theorem gives a characterization of the Fielder companion matrices in terms of unit lower Hesenberg matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' [4, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='4] If p(x) = xn + cn−1xn−1 + · · · + c1x + c0 is a polynomial over R with n ≥ 2, then F is an n × n Fiedler companion matrix to p(x) if and only if F is equivalent to a unit lower Hessenberg matrix as in (1) with the additional property that if −ck is in position (i, j) then −ck+1 is in position (i − 1, j) or (i, j + 1) for 1 ≤ k ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' An alternative way to describe the unit lower Hesenberg matrix in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='1 is to say that the variable entries of R in (1) form a lattice-path from the bottom-left corner to the top-right corner of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The first two matrices in Figure 1 are examples of Fiedler companion matrices since the variable entries of R form a lattice-path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The last matrix in Figure 1 is not a Fiedler companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' If F is a Fiedler companion matrix, the initial step size of F is the number of coefficients other than c0 in the row or column containing both c0 and c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The first matrix in Figure 1 has initial step size three and the second matrix in Figure 1 has initial step size one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES 5 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Note that equivalent matrices have the same initial step size since transpo- sitions and permutation equivalence does not change the number of coefficients in the row or column containing c0 and c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Using Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2, one can describe the nonzero entries of the inverse of a Fiedler companion matrix: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be a polynomial over R with n ≥ 2 and c0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let F be a Fiedler companion matrix to p(x) with an initial step size t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Then (1) F −1 has t + 1 entries equal to − 1 c0, − c1 c0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , − ct c0, (2) F −1 has n − 1 − t entries equal to ct+1, ct+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , cn−1, (3) F −1 has n − 1 entries equal to 1, and (4) the remaining entries of F −1 are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Since F is a companion matrix to p(x), by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='1, the matrix F is equivalent to a lower Hessenberg matrix C of the form (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Since F and C are equivalent, it follows that the matrices F −1 and C−1 are equivalent, so it suffices to show that the matrix C−1 satisfies conditions (1) − (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Since F is a Fielder companion matrix, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='1 implies that c1 is either directly above c0 in C or directly to the right of c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' If c1 is to right of c0 in C, then all other entries in the column containing c0 is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Alternatively, if c1 is above c0, all entries to the right of c0 in C are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2, which gives us the inverse of a unit lower Hessenberg matrix, applies to the matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' By our above observation, the vector u or the vector y must be the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Without loss of generality, let yT be zero, which means that − 1 c0uyT − H = −H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' If the initial step size of A is t, then there will be t nonzero elements in u, and it will have the form u = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 0 −ct .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' −c1 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2 the inverse of the matrix C then has the form (2) C−1 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0T 0T − 1 c0 Im O 0 −H In−m−1 1 c0u \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' From (2), we can describe the entries of C−1: m + n − m − 1 = n − 1 entries are 1 (coming from the submatrices Im and In−m−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' ct+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , cn−1, which all belong to the submatrix −H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' the entry − 1 c0 from the top-right corner;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' and the entries − c1 c0 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , − ct c0 from the term 6 MICHAEL COX, KEVIN N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP 1 c0u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Moreover, the rest of the entries of C−1 are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' We have now shown that C−1, and hence F −1, has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='3 mimics [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' As observed in [7], the initial step size of a Fiedler companion matrix is equal to the number of initial consecutions or inversions of the permuation associated with the Fielder companion matrix, as defined in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' We can now compute the condition number for any Fiedler companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' This result first appeared in [3], but we can avoid the formal analysis of the permutation that was used to construct the Fiedler companion matrix, as well as the associated concepts of consecution and inversion of a permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' [3, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='1] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be a polynomial over R with n ≥ 2 and c0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let F be a Fiedler companion matrix to p(x) with an initial step size t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Then κ(F)2 = ||F||2 · � (n − 1) + 1 + |c1|2 + · · · + |ct|2 |c0|2 + |ct+1|2 + · · · + |cn−1|2 � , with ||F||2 = (n − 1) + |c0|2 + |c1|2 + · · · + |cn−1|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' This result follows from the fact that F is a unit sparse companion matrix (so it contains n − 1 entries equal to 1 and the entries −c0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , −cn−1), and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='3, which describes the entries of F −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' □ Because the condition number κ(F) of a Fiedler companion matrix F depends only upon the initial step size and not the permutation σ, we can derive the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' [3, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='3] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be a polynomial over R with n ≥ 2 and c0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let A and B be Fiedler companion matrices to the polynomial p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' If the initial step size of both A and B is t, then κ(A) = κ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Since condition numbers of Fiedler companion matrices depend on the initial step size, let St = {F | F is a Fiedler companion matrix to p(x) with initial step size t}, and define κ(t) = κ(F) for F ∈ St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' We can now recover a result of [3] that allows us to compare the condition numbers of Fielder matrices while again avoiding any reference to the permutation σ used to define a Fiedler matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' [3, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='5] Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be a polynomial over R with n ≥ 2 and c0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Then (1) if |c0| < 1, then κ(1) ≤ κ(2) ≤ · · · ≤ κ(n − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' (2) if |c0| = 1, then κ(1) = κ(2) = · · · = κ(n − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' and (3) if |c0| > 1, then κ(1) ≥ κ(2) ≥ · · · ≥ κ(n − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Note that by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='6, κ(A) is the same for all A ∈ St, so κ(t) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The conclusions follow from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' □ CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES 7 One of our new results is to compare the condition number of a Fielder companion matrix of p(x) to the condition number of other companion matrices of p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In particular, if a Fiedler companion matrix F has a smaller condition number than another companion matrix C to the same polynomial p(x), then the ratio κ(C) κ(F ) can be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' This result is similar in spirit to [3, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let p(x) = xn + cn−1xn−1 + · · · + c1x + c0 be a polynomial over R with n ≥ 2, and c0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let F be a Fielder companion matrix to p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Further, suppose C is any companion matrix to p(x) whose lower Hessenberg form is C = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 Im O uC HC In−m−1 −c0 yT C 0T \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb such that either uC or yT C is the zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' If κ(F) ≤ κ(C), then 1 ≤ κ(C) κ(F) ≤ κ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The conclusion that 1 ≤ κ(C) κ(F ) is immediate from the hypothesis that κ(F) ≤ κ(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='1 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='4, we can assume F is in unit lower Hessenberg form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' As such, let F = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 Il O uF HF In−l−1 −c0 yT F 0T \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' and let t be the initial step size of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' We want to show that ||C|| · ||C−1|| ||F|| · ||F −1|| ≤ ||F|| · ||F −1||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Since C and F are unit sparse companion matrices, ||C|| = ||F||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' It suffices to show that ||C−1|| ≤ ||F|| · ||F −1||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Using equivalence, we may assume without loss of generality that uC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2, C−1 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 1 c0yT C 0T − 1 c0 Im O 0 −HC In−m−1 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' since uC = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Then (3) ||C−1||2 = (n − 1) + � 1 c0 �2 + � ci∈yT C ���� ci c0 ���� 2 + � ck∈HC |ck|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 8 MICHAEL COX, KEVIN N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP where c ∈ H (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' c ∈ y) means −c is an entry in H (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' On the other hand, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='3, (4) ||F||2 · ||F −1||4 = � (n − 1) + n−1 � i=0 |ci|2 � \uf8ee \uf8f0(n − 1) + � 1 c0 �2 + t � i=1 ���� ci c0 ���� 2 + n−1 � j=t+1 |cj|2 \uf8f9 \uf8fb 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' We want to show that ||C−1|| ≤ ||F||·||F −1||2 which is equivalent to showing that ||C−1||2 ≤ ||F||2 · ||F −1||4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' To do this, for each of the four different summands in (3), we show that there exists distinct terms in ||F||2 ·||F −1||4 that are greater than or equal to the summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Here we rely on the fact that there are no negative summands in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Partially expanding out (4), we have ||F||2 · ||F −1||4 = (n − 1)3 + (n − 1)2 � 1 c0 �2 + (n − 1) �n−1 � i=0 |ci|2 � � 1 c0 �2 + (n − 1)2 n−1 � j=0 |cj|2 + other non-negative terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Consequently, ||C−1||2 = (n − 1) + � 1 c0 �2 + � ci∈yT C ���� ci c0 ���� 2 + � ck∈HC |ck|2 ≤ (n − 1)3 + (n − 1)2 � 1 c0 �2 + (n − 1) �n−1 � i=0 |ci|2 � � 1 c0 �2 + (n − 1)2 n−1 � j=0 |cj|2 ≤ ||F||2 · ||F −1||4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Striped Companion Matrices In this section we explore a particular class of companion matrices known as striped companion matrices, which were introduced in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' A striped companion matrix to a polynomial p(x) = xn + cn−1xn−1 + · · · + c1x + c0 has the property that the coefficients −c0, −c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , −cn−1 form horizontal stripes in the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In particular, if t = (t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' tr) is an ordered r-tuple of positive integers with t1 +t2 +· · ·+tr = n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' and t1 ≥ ti for 2 ≤ i ≤ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' then we define the striped companion matrix Sn(t) to be the companion matrix of unit Hes- senberg form Sn(t) = \uf8ee \uf8f0 0 It1−1 O R In−t1 0T \uf8f9 \uf8fb (5) with the (n − t1 + 1) × t1 matrix R having r nonzero rows and with the ith nonzero row of R having ti variables in the first ti positions and ti − 1 zero rows immediately above it in CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES 9 R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' for 1 < i ≤ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Note that this implies the first row of R is a nonzero row with t1 leading nonzero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' For example, S7(3, 2, 2) = � ���������� 0 1 0 0 0 0 0 0 0 1 0 0 0 0 −c4 −c5 −c6 1 0 0 0 0 0 0 0 1 0 0 −c2 −c3 0 0 0 1 0 0 0 0 0 0 0 1 −c0 −c1 0 0 0 0 0 � ���������� , and S8(3, 3, 2) = � ������������ 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 −c5 −c6 −c7 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 −c2 −c3 −c4 0 0 0 1 0 0 0 0 0 0 0 0 1 −c0 −c1 0 0 0 0 0 0 � ������������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' As the next theorem shows, in some cases the stripped companion matrices can have a better condition number than a Fielder companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Suppose n = k(m + 1) for some positive k, m ∈ Z and p(x) = xn + cn−1xn−1 + · · · + c1x + c0 with c0 = 1, c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , cn−1 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' There exists a striped companion matrix S = Sn(k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , k) for p(x) such that κ(S) ≤ κ(F) for every Fiedler companion matrix F if and only if (6) m � j=1 �k−1 � i=1 |cicjk − cjk+i|2 � ≤ m � j=1 �k−1 � i=1 |cjk+i|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let S = Sk(m+1)(k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , k), and let F be a Fiedler companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Since ||S|| = ||F|| as noted in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='3, it suffices to show that ||S−1|| ≤ ||F −1|| if and only if equation (6) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2, S−1 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 −c1 −c2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' −ck−1 0T −1 Ik−1 O 0 −c1cmk + cmk+1 −c2cmk + cmk+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' −ck−1cmk + c(m+1)k−1 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 0 −c1c2k + c2k+1 −c2c2k + c2k+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' −ck−1c2k + c3k−1 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 0 −c1ck + ck+1 −c2ck + ck+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' −ck−1ck + c2k−1 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 0 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 0 Imk −cmk 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 0 −c2k 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 0 −ck 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Thus ||S−1||2 = n + k−1 � j=1 |cj|2 + m � j=1 |cjk|2 + m � j=1 �k−1 � i=1 |cicjk − cjk+i|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 10 MICHAEL COX, KEVIN N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='5, ||F −1||2 = n + k−1 � j=1 |cj|2 + m � j=1 |cjk|2 + m � j=1 �k−1 � i=1 |cjk+i|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Therefore κ(S) ≤ κ(F) if and only if m � j=1 �k−1 � i=1 |cicjk − cjk+i|2 � ≤ m � j=1 �k−1 � i=1 |cjk+i|2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' □ We can deduce the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Suppose n = k(m + 1) for some m, k ∈ Z and p(x) = xn + cn−1xn−1 + · · + c1x + c0 with c0 = 1, c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , cn−1 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Suppose F is any Fiedler companion matrix for p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' If |cicjk − cjk+i| ≤ |cjk+i|, for 1 ≤ j ≤ m and 1 ≤ i ≤ k − 1, then there exists a striped companion matrix S = Sn(k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , k), such that κ(S) ≤ κ(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let p(x) = x9 + 8x8 + 6x7 + 2x6 + 5x5 + 8x4 + 3x3 + 3x2 + 2x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Note that the inequalities in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let F be any Fiedler companion to p(x) and consider the striped companion matrix S = S9(3, 3, 3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=', S9(3, 3, 3) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 −2 −6 −8 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 −3 −8 −5 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 −1 −2 −3 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Then ||S|| = ||F|| = √ 224, but κ(S) = √ 224 √ 63 < κ(F) = √ 224 √ 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' One extreme example of how the inequalities in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2 can be met is if c0 = 1 and the striped companion matrix in line (5) has rank(R) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In this case, the inequalities are trivially met as described in the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' A more general result can be developed for striped companion matrices with differing stripe lengths;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=', see [1, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Given p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 with c0 = 1, and c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , cn−1 ∈ R, let S be a striped companion matrix to the polynomial p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' If S = \uf8ee \uf8ef\uf8f0 0 Im O R In−m−1 0T \uf8f9 \uf8fa\uf8fb with rank(R) = 1, then κ(S) ≤ κ(F) for any Fiedler companion matrix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' This result follows from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2 by observing that |cicjk − cjk+i| = 0 for all 1 ≤ j ≤ m and 1 ≤ i ≤ k − 1, if and only if rank(R) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In particular, rank(R) = 1 if and only every 2 × 2 submatrix of R has determinant zero, which is true if and only if |cicjk − cjk+i| = 0 for 1 ≤ j ≤ m and 1 ≤ i ≤ k − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Note that we are using the fact that � −cjk −cjk+i −c0 −ci � is a 2 × 2 submatrix of R and c0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' □ Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let b, k ∈ R and consider the polynomial p(x) = x6 + (bk3)x5 + (bk2)x4 + (bk2)x3 + (bk)x2 + kx + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' If S = S6(2, 2, 2) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 1 0 0 0 0 −bk2 −bk3 1 0 0 0 0 0 0 1 0 0 −bk −bk2 0 0 1 0 0 0 0 0 0 1 −1 −k 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb and F is any Fiedler companion matrix for p(x), then �κ(F) κ(S) �2 = b2k6 + b2k4 + b2k4 + b2k2 + k2 + 6 b2k4 + b2k2 + k2 + 6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In this case, for sufficiently large k, κ(F) κ(S) ≈ k demonstrating a significantly better condition number for S compared to any Fiedler com- panion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' As shown in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='4, if the rank of the submatrix R in the striped companion matrix S has rank(R) = 1, then the inequality κ(S) ≤ κ(F) holds for any Fiedler companion matrix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Note that in the striped companion matrix given in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='5, the corresponding submatrix R has rank one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Observe also that we can write p(x) has p(x) = q(x) + (bk)x2q(x) + (bk2)x4q(x) + x6 with q(x) = 1 + kx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' This generalizes: if the matrix S in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='4 has rank(R) = 1, then p(x) = xn+q(x)f(x) for some polynomial q(x) with deg(q(x)) = m and deg(f(x)) = n − m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Moreover, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='4 can be improved by giving an estimate on κ(F ) κ(S) for any Fiedler companion matrix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Suppose n = k(m + 1) and p(x) = q(x) + b1xkq(x) + b2x2kq(x) + · · · + bmxmkq(x) + x(m+1)k with q(x) = ak−1xk−1 + ak−2xk−2 + · · · + a1x + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let S = Sn(k, k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , k) and F be any Fiedler companion matrix to p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' If (b2 1 + · · · + b2 m) is sufficiently large, then �κ(F) κ(S) �2 ≈ (a2 1 + · · · + a2 k−1 + 1), 12 MICHAEL COX, KEVIN N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP or if (a2 1 + · · · + a2 k−1) is sufficiently large, then �κ(F) κ(S) �2 ≈ (1 + b2 1 + · · · + b2 m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' By Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='3, κ(F ) κ(S) = ||F −1|| ||S−1||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2, ||S−1||2 = a2 1 + · · · + a2 k−1 + b2 1 + · · · + b2 m + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='5 we can determine that ||F −1||2 = (1 + b2 1 + · · · + b2 m)(a2 1 + · · · + a2 k−1) + (b2 1 + · · · + b2 m) + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Therefore, �κ(F) κ(S) �2 = (1 + b2 1 + · · · + b2 m)(a2 1 + · · · + a2 k−1) + (b2 1 + · · · + b2 m) + n (a2 1 + · · · + a2 k−1) + (b2 1 + · · · + b2m) + n and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Generalized companion matrices: a case study In the previous sections, we focused on the condition numbers of unit sparse companion matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In this section, we initiate an investigation into the condition numbers of a family of matrices that are not companion matrices, but have properties similar to companion matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' To date, there appears to be little work done on this approach, so the work in this section can be seen as providing a proof-of-concept for future projects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' These results can also be viewed in the broader context of developing the properties of generalized companion matrices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=', see [4, 6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Roughly speaking, given a polynomial p(x) = xn + cn−1xn−1 + · ·+c1x1 +c0, a generalized companion matrix A is a matrix whose entries are polynomials in the c0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , cn and whose characteristic polynomial is p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' See [6] for more explicit detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Instead of considering the general case, we focus on a particular family of matrices and their condition numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' This case study shows that the condition numbers can improve on those of Frobenius (or Fiedler) companion matrices under some extra hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' We now define our special family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let p(x) = xn+cn−1xn−1+· · ·+c1x+c0 be a polynomial over R with n ≥ 2 and let a ∈ R be any real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Fix an integer ℓ ∈ {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , n − 2} and let aT = (−cn−1, −cn−2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , −cℓ+1) and bT = (−cℓ−2, −cℓ−3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , −c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Then let (7) Mn(a, ℓ) = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 a In−ℓ−1 O O −cℓ + a W I2 O −cℓ−1 + acn−1 b O O Iℓ−2 −c0 O O O \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES 13 \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 −c6 1 0 0 0 0 0 −c5 0 1 0 0 0 0 −c4 + a 0 0 1 0 0 0 −c3 + ac6 −a 0 0 1 0 0 −c2 0 0 0 0 1 0 −c1 0 0 0 0 0 1 −c0 0 0 0 0 0 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The matrix M7(a, 4) where W is a 2 × (n − ℓ − 1) matrix having W2,1 = −a and zeroes in every other entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Informally, the matrix Mn(a, ℓ) is constructed by starting with the Frobenius companion matrix which has all the coefficents of p(x) in the first column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Then we fix a row that is neither the top row nor one of the bottom two rows (this corresponds to picking the ℓ), and then adding a to cℓ in the (n − ℓ)-th row, and −a in the column to the right and one below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' We then also add acn−1 to the first entry in the (n − ℓ + 1)-th row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Note that when a = 0, Mn(0, ℓ) is equivalent to the Frobenius companion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' We can thus view Mn(a, ℓ) as a perturbation of the Frobenius companion matrix when a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' As an example, the matrix M7(a, 4) is given in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' We wish to compare the condition number of Mn(a, ℓ) with the Frobenius (and Fiedler) companion matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In some cases our new matrix Mn(a, ℓ) can provide us with a smaller condition number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The next lemma gives the inverse of Mn(a, ℓ) and shows that the char- acteristic polynomial of Mn(a, ℓ) is p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be a polynomial over R, with n ≥ 2 and c0 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let a ∈ R and ℓ ∈ {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , n − 2}, and let M = Mn(a, ℓ) be constructed from p(x) as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Then (i) the characteristic polynomial of M is p(x), and (ii) if c0 ̸= 0, then M−1 = 1 c0 \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0T 0T 0T −1 c0In−ℓ O O a −c0W c0I2 O −cℓ + a −cℓ−1 O O c0Iℓ−2 b \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' (i) We employ the fact that the determinant of a matrix is a linear function of its rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In particular, if M = Mn(a, ℓ), we observe that row n − ℓ of xIn − M can be written 14 MICHAEL COX, KEVIN N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' VANDER MEULEN, ADAM VAN TUYL, AND JOSEPH VOSKAMP as u + av for some vectors u and v such that u is not a function of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Row n − ℓ + 1 of xIn − M can also be written in a similar manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let k = n − ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Thus applying linearity to row k gives us det(xIn − M) = det \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed xIn − \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 a In−ℓ−1 O O −cℓ W I2 O −cℓ−1 + acn−1 b O O Iℓ−2 −c0 O O O \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 + a(−1)xℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' (8) Note that the term a(−1)xℓ in (8) comes from computing the determinant of the matrix A′ formed by replacing the k-th row of the matrix xIn−M with the row � −a 0 · · 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Do- ing a row expansion along the k-th row of A′, the determinant of A′ is (−1)k+1(−a)det(A′′) where A′′ is a block lower diagonal matrix with diagonal blocks D1 and D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Furthermore, D1 is a (k−1)×(k−1) lower triangular matrix with −1 on all the diagonal entries, and D2 is a ℓ × ℓ upper triangular matrix with x on all the diagonal entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' So det(A′′) = (−1)k−1xℓ, and hence det(A′) = (−1)k+1(−a)(−1)k−1xℓ = (−a)xℓ, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' We now apply linearity to row k + 1 in the matrix that appears on the right-hand side of (8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' in particular, a similar argument shows that the right-hand side (8) is equal to (9) det \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed xI − \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 a Ik−1 O O −cℓ O I2 O −cℓ−1 b O O Iℓ−2 −c0 O O O \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 +a(−1)xℓ +acn−1(−1)xℓ−1 +a(x+cn−1)xℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Note that the first summand in (9) is the characteristic polynomial of a Frobenius companion matrix of p(x), and hence is p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Thus, (9) reduces to p(x) + a(−1)xℓ + acn−1(−1)xℓ−1 + a(x + cn−1)xℓ−1 = p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' (ii) A direct multiplication will show that the given matrix is the inverse M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' □ Because both Mn(a, ℓ) and its inverse are known, we are able to compute its condition number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In the next lemma, instead of providing the general formula, we compute the condition number under the extra assumption that c0 = 1 in the polynomial p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let p(x) = xn + cn−1xn−1 + cn−2xn−2 + · · · + c1x + c0 be a polynomial over R, with n ≥ 2, and suppose that c0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let a ∈ R and ℓ ∈ {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , n − 2}, and let M = Mn(a, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Then κ(M)2 = � v + a2 + (a − cℓ)2 + (acn−1 − cℓ−1)2� � v + a2 + (a − cℓ)2 + c2 ℓ−1 + 1 � with v = n − c2 ℓ−1 − c2 ℓ + n−1 � i=1 c2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' CONDITION NUMBERS OF HESSENBERG COMPANION MATRICES 15 The next result illustrates the desired proof-of-concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In particular, the result shows that in special cases, the condition number of the matrix Mn(a, ℓ), which has properties similar to a companion matrix, has a condition number smaller than any Fielder compan- ion matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Although the scope of this result is limited, it does suggest that generalized companion matrices, and in particular perturbations of the Frobenius companion matrix, can provide better condition numbers in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let n ≥ 2, and fix ℓ ∈ {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , n − 2} and t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Set p(x) = xn + txn−1 + txℓ + t2xℓ−1 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let M = Mn(t, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Then, for any Fieldler companion matrix F of p(x), κ(F)2 κ(M)2 = (n + 2t2 + t4)2 (n + 2t2)(n + 1 + 2t2 + t4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' In particular, for t for sufficiently large, κ(F ) κ(M) ≈ 1 √ 2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2, κ(M)2 = � 1 + t2 + (n − 1) + a2 + (a − t)2 + (at − t2)2� � 1 + t2 + (n − 1) + a2 + (a − t)2 + t4 + 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Setting a = t gives κ(M)2 = (n + 2t2)(n + 1 + 2t2 + t4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' We use Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='5 to compute κ(F)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Note that since c0 = 1, κ(F) is independent of the initial step size of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Hence κ(F) = ((n − 1) + 1 + t4 + t2 + t2) = (n + 2t2 + t4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Thus we have κ(F)2 κ(M)2 = (n + 2t2 + t4)2 (n + 2t2)(n + 1 + 2t2 + t4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' The limit of the right hand side is t2 2 as t → ∞, which implies the final statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' □ The following result gives another case where we can make a matrix with smaller condi- tion number than any other Fielder companion matrix, providing additional evidence that generalized companion matrices may be of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let n ≥ 2, and fix ℓ ∈ {3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' , n−2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let p(x) = xn+cn−1xn−1+· · ·+c1x+c0 with c0 = 1, and (cℓcn−1)2 < 2cℓ−1cℓcn−1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let M = Mn(cℓ, ℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Then κ(M) < κ(F) for every Fieldler companion matrix F of p(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Let v = n − c2 ℓ − c2 ℓ−1 + �n−1 i=1 c2 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Because c0 = 1, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='5 all Fielder companion matrices F have condition number κ(F) = (v + c2 ℓ + c2 ℓ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='2, with a = cℓ, κ(M)2 = (v + c2 ℓ + (cℓcn−1 − cℓ−1)2)(v + c2 ℓ + c2 ℓ−1 + 1) = (v + c2 ℓ + c2 ℓ−1 + ((cℓcn−1)2 − 2cℓ−1cℓcn−1))(v + c2 ℓ + c2 ℓ−1 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' If we set w = (v +c2 ℓ +c2 ℓ−1), then κ(M)2 = (w−y)(w+1) with y = 2cℓ−1cℓcn−1 −(cℓcn−1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' But y > 1 by hypothesis, thus κ(M)2 < w2 = κ(F)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' □ 16 MICHAEL COX, KEVIN N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' [7] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Vander Meulen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Vanderwoerd, Bounds on polynomial roots using intercyclic companion matri- ces, Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' 539 (2018) 94–116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content=' Unit 202 - 133 Herkimer Street, Hamilton, ON, L8P 2H3, Canada Email address: michael.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='cox@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='com Department of Mathematics, Redeemer University College, Ancaster, ON, L9K 1J4, Canada Email address: kvanderm@redeemer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='ca Department of Mathematics and Statistics, McMaster University, Hamilton, ON, L8S 4L8, Canada Email address: vantuyl@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99FQT4oBgHgl3EQfJzVJ/content/2301.13257v1.pdf'} +page_content='mcmaster.' metadata={'source': 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b/EdE2T4oBgHgl3EQf9wni/content/tmp_files/2301.04232v1.pdf.txt @@ -0,0 +1,10079 @@ +Astronomy & Astrophysics manuscript no. main +©ESO 2023 +January 12, 2023 +Barium and related stars, and their white-dwarf companions +III. The masses of the white dwarfs +A. Escorza1 and R. J. De Rosa1 +European Southern Observatory, Alonso de Córdova 3107, Vitacura, Santiago, Chile +e-mail: ana.escorza@eso.org +January 12, 2023 +ABSTRACT +Context. Masses are one of the most difficult stellar properties to measure. In the case of the white-dwarf (WD) companions of Barium +(Ba) stars, the situation is worse. These stars are dim, cool, and difficult to observe via direct methods. However, Ba stars were polluted +by the Asymptotic Giant Branch (AGB) progenitors of these WDs with matter rich in heavy elements, and the properties of their WD +companions contain key information about binary interaction processes involving AGB stars and about the slow-neutron-capture(s)- +process of nucleosynthesis. +Aims. With this study, we aim to determine accurate and assumption-free masses for the WD companions of as many Ba stars as +possible. We want to provide new observational constraints that can help us learn about the formation and evolution of these post- +interaction binary systems and about the nucleosythesis processes that took place in the interiors of their AGB progenitors. +Methods. We combined archival radial-velocity data with Hipparcos and Gaia astrometry using the software package orvara, a code +designed to simultaneously fit a single Keplerian model to any combination of these types of data using a parallel-tempering Markov +chain Monte Carlo method. We adopted Gaussian priors for the Ba star masses and for the parallaxes, and assumed uninformative +priors for the orbital elements and the WD masses. +Results. We determined new orbital inclinations and companion masses for 60 Ba star systems. These results include a couple of new +orbits and several improved orbits for the longest-period systems. Additionally, we unravelled a new triple system that was not known +before and constrained the orbits and the masses of the two companions. +Conclusions. The WD mass distribution presented in this work is compatible with that of field WDs and with the distributions +published before for Ba star companions. A few WD companions have masses higher than 0.8 M⊙, considering 1-σ uncertainties. +This indicates that they might come from AGB stars that are more massive than 3 M⊙. These masses are higher than what the +abundance ratios on Ba star atmospheres and theoretical models of the s-process of nucleosynthesis seem to expect, raising interesting +questions about the formation of these systems. +Key words. white dwarfs - stars: late-type - stars: chemically peculiar - binaries: spectroscopic - astrometry - stars: evolution +1. Introduction +About half of the elements heavier than iron are synthesized by +the slow neutron capture (s-) process of nucleosynthesis (e.g. +Burbidge et al. 1957; Clayton et al. 1961; Käppeler et al. 2011). +The main astrophysical site that meets the appropriate condi- +tions for the s-process to operate is the helium-rich intershell +in the interiors of thermally pulsing Asymptotic Giant Branch +(tp-AGB) stars (e.g. Lugaro et al. 2003b; Cristallo et al. 2009; +Karakas 2010; Käppeler et al. 2011). However, the overabun- +dance of s-process elements on the surface of a star is not a +unique feature of AGB stars. Barium (Ba) stars are an example +of s-process enriched objects that have not reached the tp-AGB +phase yet. They are known to form when an AGB companion +pollutes them in a binary system (e.g. McClure et al. 1980; Mc- +Clure 1984; Udry et al. 1998a; Jorissen et al. 1998). The mass +donors in these systems evolved off the AGB long ago and are +now dim white dwarfs (WD), while the accretors – the Ba stars +– are observed on the main sequence (e.g. North & Duquennoy +1991; Jorissen & Boffin 1992; North et al. 1994, 2000; Pereira +2005; Kong et al. 2018; Escorza et al. 2019b), the red-giant (e.g. +Bidelman & Keenan 1951; McClure 1983; Udry et al. 1998b; +Jorissen 2004; Escorza et al. 2017; Jorissen et al. 2019), and the +AGB (as extrinsic S stars, e.g. Jorissen et al. 1998, 2019; Shetye +et al. 2020) phases. +Although their exact formation channel and the mass- +transfer mechanisms involved are not well understood (e.g. Tout +& Eggleton 1988; Han et al. 1995; Soker 2000; Pols et al. 2003; +Bonaˇci´c Marinovi´c et al. 2008; Izzard et al. 2010; Dermine et al. +2013; Abate et al. 2018; Saladino & Pols 2019; Gao et al. 2023), +our knowledge about the spectroscopic orbital parameters of Ba +star systems and about the stellar properties of the Ba stars them- +selves is generally well established (e.g. Escorza et al. 2019b; +Jorissen et al. 2019, and references therein). Additionally, the +evolutionary link between dwarf and giant Ba stars is well ac- +cepted (e.g. Escorza et al. 2020). However, not much is known +about the WD companions. The mass-function distribution of +Ba star systems is consistent with a narrow distribution of com- +panion masses peaking at 0.6 M⊙ (e.g. Webbink 1986; McClure +& Woodsworth 1990; Jorissen et al. 1998; Merle et al. 2016; +Jorissen et al. 2019; Escorza et al. 2019a), but very few abso- +lute masses have been determined, since there is normally no in- +formation about the orbital inclinations of these systems (a few +exceptional cases were published by Pourbaix & Jorissen 2000; +Escorza et al. 2019b; Jorissen et al. 2019, among others, by com- +bining the orbital parameters of Ba stars with Hipparcos astro- +Article number, page 1 of 49 +arXiv:2301.04232v1 [astro-ph.SR] 10 Jan 2023 + +A&A proofs: manuscript no. main +metric data). These WDs are cool, dim, and directly undetectable +in most cases; although, Böhm-Vitense et al. (1984, 2000); Gray +et al. (2011), among others detected UV excess flux attributable +to the WD in a few Ba star systems. +The masses of the WD companions of Ba stars contain im- +portant information about the AGB progenitors and the nucle- +osynthesis processes that took place in their interiors, and they +are important input for binary interaction models. Even though +mixing and dilution processes such as thermohaline mixing (e.g. +Proffitt & Michaud 1989; Charbonnel & Zahn 2007; Stancliffe +et al. 2007; Stancliffe & Glebbeek 2008; Aoki et al. 2008), ro- +tationally induced mixing (e.g. Denissenkov & Tout 2000), or +atomic diffusion (e.g. Matrozis & Stancliffe 2016, 2017) might +impact the final level of s-process abundance on Ba stars, corre- +lations between these abundances and the WD mass can give us +observational information about the efficiency of the s-process +at different masses and metallicities and help us constrain AGB +models (e.g. Cseh et al. 2022) and mass-transfer and dilution +models (e.g. Stancliffe 2021). The ratio between the amount of +heavy s-process elements (hs), such as Ba, La, or Ce, and light +s-process elements (ls), such as Sr, Y, or Zr, on the surface of Ba +stars suggests that the material accreted by these stars was syn- +thesized by low-mass AGB stars (< 3 M⊙ ; Lugaro et al. 2003a, +2012, 2016; Cseh et al. 2018; Karinkuzhi et al. 2018), which +still needs to be confirmed by measuring these WD masses. Ad- +ditionally, Jorissen et al. (2019) suggested that WD compan- +ions of strong Ba giants (based on the Ba index introduced by +Warner 1965) are more massive on average than the WD com- +panions of mild Ba stars. However, most of their masses were +determined under the assumption of a constant (or very narrow +distribution of) Q = M3 +WD/(MBa + MWD)2 as proposed by Web- +bink (1988) and McClure & Woodsworth (1990), so this trend +still needs to be confirmed with assumption-free measurements +of WD masses. +In the first two papers of this series, Jorissen et al. (2019) +and Escorza et al. (2019b) collected old and new radial-velocity +(RV) data to study the orbits of giant and dwarf Ba stars, re- +spectively. Additionally, we used spectroscopically-determined +stellar parameters and Gaia DR2 distances (Lindegren et al. +2018; Bailer-Jones et al. 2018) to locate these stars on the +Hertzsprung–Russell diagram (HRD). By comparing their loca- +tion on the HRD with STAREVOL evolutionary tracks (Siess +et al. 2000; Siess 2006, 2008) and following the methodology +described in Escorza et al. (2017), we also determined accurate +masses for the primary stars of these systems, the Ba stars. In this +third article, we focus on the faint WD companions. We used the +orvara software (Brandt et al. 2021c) to combine all the radial- +velocity data available, the astrometric measurements from the +Hipparcos mission (Perryman et al. 1997), the Gaia positions and +proper motions (Lindegren et al. 2021), and the information in +the Hipparcos-Gaia Catalogue of Accelerations (HGCA; Brandt +2018, 2021) to determine the astrometric orbital parameters of +as many Ba star systems as possible (see Sect. 2 for the descrip- +tion of the sample), and then derive the mass of the secondary +stars. All these data sets are described in Sect. 3. An important +improvement with respect to what has been attempted before +for these objects is that we use a joint astrometric-spectroscopic +model (see Sect. 4) to find new best-fitting orbital parameters in- +stead of relying only on RV data or imposing the spectroscopic +solution on the astrometric data. Our results are presented in +Sect. 5 and their implications are discussed in Sect. 6. We also +discuss the feasibility of the direct detection of the WD compan- +ion for a subset of the longest-period systems in Sect. 7. +2. Target selection +For our methodology (see Sect. 4) to be applicable, a target must +fulfil three requirements: (i) it must be part of the HGCA, (ii) +we must have a good initial estimate of the mass of the primary +star in the system, and (iii) the Hipparcos solution cannot not be +more complex than the 5-parameter solution. As a starting point, +we selected all the Ba stars from the samples studied by Joris- +sen et al. (2019), Escorza et al. (2019b) and North et al. 2020 +that have Hipparcos identifiers. We excluded confirmed triple +systems, stars whose Ba star nature was not certain or is un- +der current investigation (Escorza et al. under review), and a few +systems that had an acceleration solution or an orbital solution +in the Hipparcos data reduction (solution types, Sn, equal to 7, 9 +and 75). We ended up with 60 systems. +Table 1 presents our target list. In addition to the most com- +monly used identifier, we include the Hipparcos identifier of +each system and the Ba star type. We distinguish between pre- +RGB, which are all the stars classified as dwarfs or subgiants +by Escorza et al. (2019b) and North et al. (2020), and mBag or +sBag which are stars classified as mild or strong Ba giants by +Jorissen et al. (2019) based on their [La/Fe] and [Ce/Fe] values +(as measured by Smith 1984; Allen & Barbuy 2006a,b; Pereira +et al. 2011; Karinkuzhi & Goswami 2014, 2015; Luck 2014; de +Castro et al. 2016; Merle et al. 2016; Van der Swaelmen et al. +2017; Karinkuzhi et al. 2018; Jorissen et al. 2019) and on the Ba +index introduced by Warner (1965). The table also lists the Ba +star masses (MBa) that we used as a prior in our MCMC model +(see Sect. 4) and the metallicity of the system, both values col- +lected from Jorissen et al. (2019), Escorza et al. (2019b) or North +et al. (2020) unless explicitly specified. For this work, we recom- +puted the primary masses for the ten systems that were part of +the non-single-star (NSS) Gaia DR3 catalogues (Gaia Collabo- +ration et al. 2022). We followed the exact same procedure fol- +lowed and described in the mentioned papers and used the same +STAREVOL grids of models (Siess et al. 2000; Siess & Arnould +2008; Escorza et al. 2017), but we used the NSS Gaia DR3 par- +allaxes to recalculate their luminosities and masses. Finally, the +last column of Table 1 includes the sources where we found the +archival RV data used in our analysis. +3. Radial velocity and astrometric data +3.1. CORAVEL, HERMES and other radial-velocity data +The most important radial-velocity monitoring programs of Ba +stars were carried out with the two CORAVEL spectrome- +ters and with the HERMES high-resolution spectrograph. The +CORAVEL spectrometers (Baranne et al. 1979) were installed +on the 1-m Swiss telescope at the Haute-Provence Observatory +and on the 1.54-m Danish telescope at ESO - La Silla, while +HERMES (Raskin et al. 2011; Raskin & Van Winckel 2014) is +mounted on the 1.2-m Flemish Mercator telescope at the Obser- +vatory El Roque de Los Muchachos. +The main results of these radial-velocity programs were pub- +lished by Jorissen & Mayor (1988); Jorissen et al. (1998); Udry +et al. (1998a,b); North et al. (2000); Gorlova et al. (2013); Joris- +sen et al. (2019); Escorza et al. (2019b) among others, and the +strength of combining the two data sets, particularly for the +longest-period systems, was discussed in the last two mentioned +papers. Jorissen et al. (2019) and Escorza et al. (2019b) also de- +scribed the data reduction process for the two instruments and +the existence of a non-zero radial-velocity offset between the +data sets due to the use of a different system of standard stars. +Article number, page 2 of 49 + +A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Table 1: List of Ba star systems to which our methodology was applied. Column 1 lists the most commonly used identifiers, while +column 2 lists the Hipparcos identifiers. Column 3 lists the Ba star type, which can be preRGB for stars classified as dwarfs or +subgiants, or mBag or sBag for stars classified as mild or strong Ba giants, respectively. Column 4 lists the primary star masses and +column 5, the metallicity of the system. These values were derived or collected by Escorza et al. (2019b) or Jorissen et al. (2019) +for preRGB and giant systems, respectively, unless otherwise indicated. Finally, the last column gives the sources of the archival +RV data we used. +BD/HD +HIP +type +MBa [M⊙] +[Fe/H] +RV ref∗ +HD +HIP +type +MBa [M⊙] +[Fe/H] +RV ref∗ +-10o4311 +80356 +preRGB +0.8 ± 0.1 +−0.65 +E19 +95241 +53791 +preRGB +1.3 ± 1.1 +−0.37 +E19,M04 +-11o3853 +73444 +preRGB +0.85 ± 0.04(1) +−0.94(1) +E19 +98991 +55598 +preRGB +1.45 ± 0.08 +−0.44 +E19 +-14o2678 +43527 +mBag +3.0 ± 0.2 +0.01 +U98a +104979 +58948 +mBag +2.7 ± 0.2 +−0.26 +J19,M90 +2454 +2235 +preRGB +1.23 ± 0.07(1) +−0.29(1) +E19,M04 +107541 +60292 +sBag +1.1 ± 0.2 +−0.63 +U98b +5424 +4347 +sBag +1.3 ± 0.4 +−0.43 +U98a +107574 +60299 +preRGB +1.11 ± 0.05 +−0.80 +E19 +16458 +13055 +sBag +1.9 ± 0.1 +−0.64 +M90 +119185 +66844 +mBag +1.7 ± 0.2 +−0.42 +J19,U98a +18182 +13596 +mBag +1.8 ± 0.2 +−0.17 +J19 +121447 +68023 +sBag +1.6 ± 0.1 +−0.90 +J95 +20394 +15264 +sBag +2.0 ± 0.2 +−0.27 +G96 +123585 +69176 +preRGB +1.0 ± 0.1(0) +−0.50 +E19 +24035 +17402 +sBag +1.3 ± 0.3 +−0.23 +U98a +123949 +69290 +sBag +1.3 ± 0.3 +−0.23 +J19,U98a +27271 +20102 +mBag +2.9 ± 0.2 +−0.07 +U98b,M04 +127392 +71058 +preRGB +0.8 ± 0.3 +−0.52 +E19 +31487 +23168 +sBag +2.5 ± 0.2(2) +−0.04(2) +M90 +139195 +76425 +mBag +2.6 ± 0.1 +−0.07 +M90,G91 +34654 +25222 +preRGB +1.19 ± 0.05(0) +−0.09 +E19 +143899 +78681 +mBag +2.4 ± 0.1 +−0.29 +U98a +40430 +28265 +mBag +2.3 ± 0.2 +−0.34 +J19 +178717 +94103 +sBag +1.6 ± 0.9 +−0.52 +M90 +43389 +29740 +sBag +1.8 ± 0.4 +−0.35 +U98b +180622 +94785 +mBag +1.8 ± 0.3 +0.03 +U98b +44896 +30338 +sBag +3.0 ± 1.2(0) +−0.25 +U98b +182274 +95293 +preRGB +1.09 ± 0.05 +−0.32 +E19,M04 +49641 +32713 +sBag +2.7 ± 1.2 +−0.3 +M90 +183915 +96024 +sBag +1.8 ± 1.0 +−0.59 +J19,J98 +49841 +32831 +mBag +2.85 ± 0.10(0) +0.2 +U98b +199939 +103546 +sBag +2.7 ± 0.4(0) +−0.22 +M90,M04 +50082 +32960 +sBag +1.6 ± 0.3 +−0.32 +U98b +200063 +103722 +mBag +2.0 ± 1.3 +−0.34 +U98b +50264 +32894 +preRGB +0.9 ± 0.1(0) +−0.34 +E19 +201657 +104542 +sBag +1.8 ± 0.5 +−0.34 +U98b +51959 +33628 +mBag +1.2 ± 0.1 +−0.21 +J19 +201824 +104684 +sBag +1.7 ± 0.4 +−0.40 +G96 +53199 +34143 +mBag +2.5 ± 0.1 +−0.20 +J19,M04 +202400 +105294 +preRGB +0.98 ± 0.08(3) +−0.7(3) +N20 +58121 +35935 +mBag +2.6 ± 0.5 +−0.01 +U98b +204075 +105881 +mBag +4.5 ± 0.3 +−0.09 +M90 +58368 +36042 +mBag +2.6 ± 0.2 +0.04 +M90 +205011 +106306 +mBag +1.8 ± 0.3 +−0.26 +J98,M90 +59852 +36613 +mBag +2.5 ± 0.3 +−0.22 +U98a +207585 +107818 +preRGB +0.90 ± 0.10(0) +−0.57 +E19 +77247 +44464 +mBag +3.9 ± 0.2 +−0.13 +M90 +210946 +109747 +mBag +1.8 ± 0.5 +−0.29 +U98b,M04 +87080 +49166 +preRGB +1.38 ± 0.15(0) +−0.60 +E19 +211594 +110108 +sBag +2.0 ± 0.3 +−0.29 +U98b +88562 +50006 +sBag +1.0 ± 0.1 +−0.53 +U98a +216219 +112821 +preRGB +1.45 ± 0.1 +−0.17 +E19 +91208 +51533 +mBag +2.3 ± 0.2 +−0.16 +U98a +218356 +114155 +mBag +4.3 ± 1.1 +−0.06 +G06,U98b +92626 +52271 +sBag +3.1 ± 0.6 +−0.15 +U98b +221531 +116233 +preRGB +1.2 ± 0.1(0) +−0.30 +E19 +95193 +53717 +mBag +2.7 ± 0.1 +−0.04 +U98a +224621 +118266 +preRGB +0.85 ± 0.06(0) +−0.4 +N20 +∗ RV reference abbreviations: E19: Escorza et al. (2019b), U98a: Udry et al. (1998a), M04: Moultaka et al. (2004), M90: McClure +& Woodsworth (1990), J19: Jorissen et al. (2019), G96: Griffin et al. (1996), U98b: Udry et al. (1998b), J98: Jorissen et al. (1998), +J95: Jorissen et al. (1995), G91: Griffin (1991), N20: North et al. (2020), G06: Griffin (2006) +Mass & metallicity references: (0) This work; (1) Bensby & Lind (2018); (2) Karinkuzhi et al. (2018); (3) North et al. (2020) +This zero-point offset depends on the stellar velocity and on the +target’s colour B-V, and there is no real consensus about how to +treat it. Jorissen et al. (2019) derived it after fitting each orbit +by minimizing the orbital residuals, while Escorza et al. (2019b) +determined a relation between the offset and B-V by comparing +old and reprocessed CORAVEL data and calculated a fixed off- +set for each studied Ba star. For this work, we combined the two +approaches. Where the RV data of a specific instrument spanned +over a full orbit or more, we treated the offset as an additional +free parameter that was optimized during the orbital fitting pro- +cess. However, for systems with very few HERMES points or +for some very long orbits, the offsets from Jorissen et al. (2019) +or Escorza et al. (2019b) were adopted and fixed. This will be +clearly indicated in the captions of each RV fit shown in Ap- +pendix A. Future monitoring with HERMES would remove the +need for this assumption, allowing us to fit the offset term di- +rectly. +To complement the main CORAVEL and HERMES data, +we collected additional radial-velocity measurements from other +works and instruments, and the sources are listed in Table 1. An +optimizable RV offset, such as the one described above between +CORAVEL and HERMES, was considered for each data set. +3.2. Hipparcos astrometric data +The Hipparcos satellite ESA (1997), launched in 1989, was the +first space mission with precision astrometry as its main goal. +Between 1989 and 1993, Hipparcos measured the location and +motion on the sky of more than 100,000 stars many times, to fig- +ure out their astrometric path. For each target in Table 1, we used +the positions and the proper motions from the Hipparcos Cata- +Article number, page 3 of 49 + +A&A proofs: manuscript no. main +logue (Perryman et al. 1997). Additionally, we also queried the +individual astrometric measurements from the re-reduction of +the Hipparcos intermediate astrometric data (IAD; van Leeuwen +2007). The coordinates are expressed in the International Celes- +tial Reference Frame (ICRF) at the 1991.25 epoch. +Since the code we are using is not yet prepared to deal with +Hipparcos solutions more complex than the 5-parameter solu- +tions, we excluded a few targets with acceleration or orbital so- +lutions from the initial sample. Some of our remaining targets +have a stochastic Hipparcos solution (Sn = 1). These represent +cases where the residuals are significantly larger than expected, +but since the proper motions and the IAD were obtained using a +5-parameter solution, we included them and gave them no spe- +cial treatment. +3.3. Gaia astrometric data +The Gaia mission (Gaia Collaboration et al. 2016, 2018, 2021) +was launched in 2013 as a successor of Hipparcos. For now, none +of the Gaia Data Releases (DR) published individual astrometric +measurements, so we queried the positions and proper motions +published for our targets in the Early DR3 catalogue (Lindegren +et al. 2021). In contrast with the Hipparcos data, these are ex- +pressed in the ICRF at the 2016 epoch. The Gaia EDR3 paral- +laxes were also queried and used as prior in the fit (see Sect. +4). Finally, in order to use an equivalent to epoch astrometry, +we also used the Gaia Observation Forecast Tool (GOST1). The +GOST provides the predicted observations and scan angles for +any Gaia source. We note that not all the planned observations +will be used in the final astrometric solution, since some pre- +dicted scans might correspond to satellite dead times or might be +unusable or rejected as outliers. For example, up to 20% of the +observations predicted by GOST were excluded from the analy- +sis published in Gaia DR2 (Brandt et al. 2021b). +Ten of the 60 targets presented in this study had a non- +single-star (NSS) solution in Gaia DR3 (Gaia Collaboration et al. +2022). These targets are: HD 50264, HD 207585, HD 221531, +HD 34654, HD 49841, HD 199939, HD 224621 and HD 87080, +which had a non-single-star solution compatible with a com- +bined astrometric and single lined spectroscopic model, and +HD 44896 and HD 123585, which had a solution compatible +with an astrometric binary. For these targets, we used the Gaia +DR3 NSS parallax as priors, instead of the EDR3 value. Even +though the Gaia DR3 NSS catalogue provided orbital inclina- +tions for these 10 systems, we decided not to include an incli- +nation prior in our calculations to first, treat all systems equally, +and second, compare our independently determined inclinations +with the new Gaia ones and validate our method. +3.4. The Hipparcos-Gaia Catalogue of Accelerations. +As an additional astrometric constraint, we used the difference +in Hipparcos and Gaia proper motions via the Hipparcos-Gaia +Catalogue of Accelerations (HGCA; Brandt 2018, 2021). This +catalogue puts the Hipparcos, Gaia, and Hipparcos-Gaia (H-G) +proper motions into the same reference frame to make them suit- +able for orbital fitting. The Hipparcos-Gaia proper motion is de- +rived from the right ascension and declination measurements +from the two missions and is by far the most precise due to +the long time elapsed between them (proper motion uncertain- +ties scale inversely with the time baseline). This acceleration in +the inertial frame can be used to improve the dynamical parame- +1 https://gaia.esac.esa.int/gost/ +ters of the companion and to measure its mass because it breaks +the mass-inclination degeneracy that RV data suffers from. We +used the EDR3 version of the HGCA (Brandt 2021) for all our +targets. +The EDR3 version of the HGCA also provides a χ2 value be- +tween the two most precise proper motion measurements (nor- +mally EDR3 and H-G). This value is meant to find accelerating +candidates for follow-up and if it is higher than ∼11.8 (Brandt +2021), the measured acceleration is considered significant and +statistically different, by 3σ, from constant proper motion. In +our case, since all our targets are known binaries, we do not +need this χ2 value to detect accelerators, but it can give us a +hint about which systems are truly benefiting from the HGCA +measurement. The queried HGCA χ2 values are included in the +last column of our result table (Table 2). +4. Orbital analysis with orvara +Orvara, developed by Brandt et al. (2021c), is designed to si- +multaneously fit a single Keplerian model to any combination +of radial velocity data and relative and absolute astrometry. The +combination of these different data sets, using Orvara or not, has +recently proven to be very powerful to improve the accuracy of +orbits and to measure precise companion masses, even for very +long period systems where the observations only cover part of +the orbit (e.g. De Rosa et al. 2020; Kervella et al. 2020; Brandt +et al. 2021c; Venner et al. 2021; Franson et al. 2022; Brandt et al. +2021a; Leclerc et al. 2022). +Orvara integrates the Hipparcos and Gaia intermediate as- +trometry package (htof; Brandt et al. 2021b) to fit the Hipparcos +epoch astrometry and the times and scan angles of individual +Gaia epochs. The code uses a parallel-tempering Markov chain +Monte Carlo method (ptmcmc, Foreman-Mackey et al. 2013) and +first fits the RV data. Orvara allows RV points from each instru- +ment to have a different RV zero point, which we need at least +for the CORAVEL-HERMES combination as discussed in Sect. +3.1. Then the absolute astrometry is included and fit for the five +astrometric parameters (positions, α and δ, proper motions, µα +and µδ, and parallax, ϖ) using htof at each MCMC step. On +top of the five astrometric parameters, we fitted the six Keple- +rian orbital elements (semimajor axis, a, eccentricity, e, time of +periastron passage, T0, argument of periastron, ω, orbital incli- +nation, i, and longitude of the ascending node, Ω), the masses of +the two components (MBa and MWD), and a radial-velocity jitter +per instrument to be added to the uncertainties. Note that while +the difference between the Hipparcos and Gaia reference frames +is taken into account in the HGCA, this is not the case for the +IAD. However, the rotation difference in the proper motions is +w = (−0.120, 0.173, 0.090) mas/yr (Fabricius et al. 2021). These +values are very small compared to the amplitudes of the proper +motion curves that we are measuring (of the order from a few to +a couple of tens mas/yr, see Appendix A), and smaller than the +residuals to these fits in most cases, so we did not take them into +account. +For this work, we assumed uninformative priors for the or- +bital elements and for the WD mass, but we adopted Gaussian +priors for the primary mass and for the parallax. For each target, +we used the MBa value given in Table 1 but using three times +the error bar as sigma to be conservative and take into account +systematic errors not accounted for in the statistical uncertainty. +Concerning the parallax, the Gaia EDR3 value was used as prior +for most targets, and the Gaia DR3 NSS parallax was used for +the 10 targets with a NSS solution. We used 15 temperatures and +for each temperature we use 100 walkers with 100,000 steps per +Article number, page 4 of 49 + +A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +walker. In a few cases, we needed to run twice as long or repeat +the calculations using an educated starting position based on our +knowledge about the systems from the RV-only fits published by +Jorissen et al. (2019) or Escorza et al. (2019b), however, in most +cases, the MCMC chains converged quite quickly. We discarded +the first 300 recorded steps (the first 15000 overall, as we saved +every 50) as the burn-in phase to produce the results presented +in Sect. 5. +For more details about the computational implementation in +orvara and htof and for case studies showing the performance +of the code, we refer to the mentioned publications. +5. Results +Table 2 lists the obtained astrometric-spectroscopic orbital pa- +rameters, the best-fitting WD masses, the χ2 of the best fit, and +the HGCA χ2 values discussed in Sect. 3.4. To make the ta- +ble easier to read, we assume that the error bars we obtained +from the MCMC fit are symmetric and listed only the largest +value. This means that in some cases, the table lists an overes- +timated uncertainty in one of the two directions. The χ2 values +are an overall absolute astrometric χ2, computed adding the χ2 +for the Hipparcos proper motions (χ2 +H), the χ2 for the long-term +Hipparcos-Gaia proper motions (χ2 +HG), and the χ2 for the Gaia +proper motions (χ2 +G). orvara uses RV jitter terms such that the +reduced χ2 of the RV fit is 1, so we did not take it into account +to evaluate the goodness of the fit. +The table is ordered based on the orbital period, with the +systems with the longest periods first. This way, we can notice +that all the systems with periods longer than ∼ 3 years have sig- +nificant astrometric accelerations according to their HGCA χ2 +values, while most of the systems below that threshold do not. +Finding such a clear threshold in a sample of confirmed binaries +is an indication of the type of systems that the HGCA can help +identify. +In addition to the table and in order to illustrate and dis- +cuss how the results that we get from orvara look like, we +include the results for the main-sequence Ba star HD 2454 in +Fig. 1. HD 2454 was first identified as a Ba dwarf by Tomkin +et al. (1989), and North et al. (2000) confirmed its binarity even +though they did not have enough data to estimate its orbital pe- +riod. More recently, Gray et al. (2011) found direct evidence of +the presence of a WD companion in the system thanks to the +Galaxy Evolution Explorer (GALEX; Martin et al. 2005) UV ob- +servations and, since 2011, HD 2454 has been part of the long- +period binary monitoring program carried out with the HER- +MES spectrograph (see Sect. 3.1). In spite of having almost three +decades of RV data between the CORAVEL and HERMES mea- +surements, Escorza et al. (2019b) were not able to constrain the +orbit either. However, combining all these RV data points with +the Hipparcos and Gaia information, we can finally estimate the +orbital elements of HD 2454 as well as the mass of its WD com- +panion. +Fig. 1 shows, on the top left panel, the astrometric orbit of +HD 2454, including the predicted position of the companion on +the scheduled date of Gaia DR3. The best-fitting orbit is plot- +ted as a black thick line, while 40 other well-fitting orbits are +colour-coded as a function of the companion mass. On the top +right panel, we show the RV curve of HD 2454. For this target +we had CORAVEL (orange circles), SOPHIE (pink diamonds), +and HERMES (green triangles) RV data. The plot shows that +leaving the RV offsets between instruments completely free pro- +duces families of solutions with similar orbits and masses but +different RV offsets (displaced vertically in the RV plot). This is +especially noticeable in cases like this one, where no data sets +covers even half an orbit. We want to note that even though we +left the RV offsets free in most cases, we always made sure that +the best-fitting solution required reasonable values and, espe- +cially in the CORAVEL-HERMES case, that these values were +close to the values obtained by Jorissen et al. (2019) and Escorza +et al. (2019b). +The two bottom panels of Fig. 1 show the fit to the proper +motions in the right ascension (left) and declination (right) direc- +tions, as measured by Hipparcos (squared data point) and Gaia +(circular data point). All the data sets included in the figures were +fitted at the same time, and the plotted models are the same in all +plots. Finally, Fig. 2 shows the one and two-dimensional projec- +tions of the posterior probability distributions of the masses of +the two components in the system and a few orbital parameters +(semi-major axis, eccentricity, and inclination) from the joint RV +and astrometric MCMC computations. This corner plot shows +that the two masses are correlated, and that the semimajor axis +is also correlated with the total mass of the system. These corre- +lations are even stronger for other targets. +We have included in Appendix A figures similar to Figs. 1 +and 2 for all the targets in our sample. Additionally, an individual +case of study of a Ba dwarf using the same method was presented +in Escorza & De Rosa (2022). +5.1. Spectroscopic orbital parameters +Even though the main goal of this work was deriving the masses +of the WD companions of all these Ba stars, an important addi- +tional result of this new method are the new orbits of HD 2454 +and BD-11o3853, which could not be constrained before, as well +as the improved orbits of a few other long-period systems. When +comparing the orbital periods obtained using orvara to those +presented in Jorissen et al. (2019), Escorza et al. (2019b) and +North et al. (2020), which were obtained by fitting only the RV +data, we get a very tight relation. The purely spectroscopic pa- +rameters and the new parameters are consistent with each other +within error bars in almost all cases, and we discuss the excep- +tions below. +5.1.1. HD 218356 +Our first orbital fit for this system converged to a period of more +than 40 years, while the period published by Griffin (2006) and +Jorissen et al. (2019) for HD 218356 was 111 days. No third ob- +ject has been detected in this system in the past, but the mild +s-process enhancement in the visible star has been flagged as +surprising given the close orbit. We performed a three-body fit, +setting strong constraints on the inner orbit using the published +spectroscopic parameters, and we succeeded to recover the or- +bital parameters of two companions, confirming that HD 218356 +is actually a triple system with a third companion in a much +longer orbit than the published period. The orbital parameters +of the system are included in Table 3 and the combined RV fit +can be seen in Fig. 3. In order to test the significance of this +detection, we compared the Akaike Information Criterion (AIC) +of the two- and three-component models using the radvel pack- +age (Fulton et al. 2018). We found a ∆AIC of 439 favouring the +three-component model. Given the masses of the two compan- +ions, we expect the WD that polluted the Ba star to be in the +outer orbit. This would also explain the mild s-process enhance- +ment reported for HD 218356. We included the corner plots with +Article number, page 5 of 49 + +A&A proofs: manuscript no. main +Table 2: Orbital elements and WD masses derived following the method described in Sect. 4 and listed in order of decreasing periods. The columns list, in order, the most +commonly used identifier, the orbital period P, the eccentricity e, the time of periastron passage, the absolute semimajor axis of the orbit a, the argument of periastron of the +visible star ω∗, the longitudes of the ascending node Ω, the orbital inclination i, and the WD companion mass MWD. To keep the table cleaner, we assumed symmetric error bars, +and included only the largest one of the two. The last two columns include the χ2 of the best fitting model and the HGCA χ2 value discussed in Sect. 3.4. +Star ID +P [days] +e +T0 [HJD] +a [AU] +ω∗ [◦] +Ω [◦] +i [◦] +MWD +Fit χ2 +χ2 +HGCA +HD 2454 +29220 ± 7670 +0.59 ± 0.04 +2458626 ± 173 +22 ± 4 +313 ± 19 +11 ± 165 +34 ± 6 +0.50 ± 0.09 +1.02 +53537 +HD 119185 +25385 ± 5114 +0.61 ± 0.08 +2477092 ± 5626 +22.5 ± 3.0 +105 ± 7 +136 ± 6 +98 ± 13 +0.65 ± 0.08 +2.82 +869 +BD-11o3853 +23376 ± 9862 +0.46 ± 0.05 +2472263 ± 10323 +19 ± 6 +199 ± 19 +133 ± 10 +102 ± 9 +0.76 ± 0.14 +0.46 +2534 +HD 104979 +18518 ± 1205 +0.12 ± 0.04 +2460663 ± 500 +21.1 ± 1.7 +180 ± 17 +34 ± 2 +147.8 ± 1.6 +0.94 ± 0.14 +0.087 +5851 +HD 218356∗ +15194 ± 2630 +0.39 ± 0.13 +2469014 ± 2835 +22 ± 4 +73 ± 24 +153 ± 17 +157 ± 5 +0.85 ± 0.25 +0.57 +388 +HD 51959 +11195 ± 475 +0.30 ± 0.05 +2459598 ± 329 +11.8 ± 1.0 +31 ± 11 +81 ± 3 +163.2 ± 0.8 +0.51 ± 0.08 +0.32 +11044 +HD 123949 +8544 ± 12 +0.9167 ± 0.0007 +2457772 ± 1 +10.8 ± 0.9 +97.1 ± 0.4 +60 ± 30 +122 ± 72 +0.78 ± 0.15 +0.52 +418 +HD 182274 +8393 ± 51 +0.039 ± 0.013 +2459883 ± 867 +9.6 ± 0.5 +67 ± 278 +68.2 ± 0.8 +50.5 ± 1.0 +0.55 ± 0.06 +0.85 +65451 +HD 18182 +8258 ± 300 +0.35 ± 0.35 +2461364 ± 3518 +10.9 ± 1.5 +194 ± 85 +148 ± 46 +33 ± 26 +0.35 ± 0.35 +1.46 +179 +HD 53199 +8233 ± 175 +0.255 ± 0.010 +2456826 ± 34 +11.7 ± 0.5 +66 ± 2 +16 ± 3 +103 ± 7 +0.64 ± 0.05 +1.51 +1873 +HD 40430 +6147 ± 278 +0.26 ± 0.02 +2457340 ± 54 +9.4 ± 0.8 +74 ± 5 +177 ± 2 +23.4 ± 0.6 +0.70 ± 0.12 +3.68 +4774 +HD 95241 +5344 ± 55 +0.807 ± 0.004 +2455739.7 ± 1.4 +7.4 ± 1.5 +107.8 ± 0.7 +131.6 ± 1.4 +108 ± 2 +0.34 ± 0.12 +4.61 +7633 +HD 139195 +5296 ± 14 +0.32 ± 0.02 +2460005 ± 78 +8.8 ± 0.3 +5.3 ± 3.5 +29 ± 3 +97.6 ± 1.1 +0.66 ± 0.05 +0.27 +421 +BD-10o4311 +4872 ± 14 +0.047 ± 0.006 +2456076 ± 105 +6.2 ± 0.7 +159 ± 8 +53.5 ± 1.2 +73 ± 3 +0.52 ± 0.11 +1.39 +20039 +HD 183915 +4344 ± 20 +0.41 ± 0.04 +2457609 ± 62 +7.1 ± 1.2 +118 ± 7 +77 ± 3 +174.3 ± 0.3 +0.61 ± 0.19 +0.18 +5410 +HD 180622 +4045 ± 30 +0.08 ± 0.04 +2458549 ± 3149 +6.9 ± 1.2 +293 ± 42 +166 ± 10 +100 ± 6 +0.80 ± 0.25 +1.03 +1948 +HD 216219 +3948 ± 23 +0.085 ± 0.050 +2456527 ± 275 +6.2 ± 0.4 +59 ± 26 +52 ± 125 +33.5 ± 1.8 +0.63 ± 0.08 +0.73 +1059 +HD 107541 +3583 ± 47 +0.095 ± 0.035 +2458478 ± 3180 +5.5 ± 0.8 +220 ± 16 +9.2 ± 7.2 +128 ± 3 +0.55 ± 0.16 +0.092 +6173 +BD-14o2678 +3481 ± 205 +0.25 ± 0.05 +2455846 ± 328 +7.0 ± 0.5 +283 ± 15 +144 ± 11 +93 ± 18 +0.67 ± 0.10 +0.28 +399 +HD 59852 +3477 ± 80 +0.14 ± 0.08 +2457280 ± 351 +6.6 ± 0.8 +95 ± 40 +139 ± 129 +26 ± 4 +0.62 ± 0.15 +3.52 +1772 +HD 201824 +2922 ± 23 +0.30 ± 0.04 +2456263 ± 102 +5.5 ± 1.1 +62 ± 9 +143 ± 78 +59 ± 66 +0.78 ± 0.28 +3.08 +765 +HD 178717 +2912 ± 14 +0.46 ± 0.03 +2455886 ± 57 +5.2 ± 0.8 +265 ± 5 +20 ± 5 +35 ± 4 +0.53 ± 0.16 +0.13 +1357 +HD 50082 +2883 ± 6 +0.19 ± 0.02 +2457496 ± 37 +5.1 ± 0.9 +207 ± 6 +22 ± 7 +63 ± 3 +0.56 ± 0.18 +1.11 +291 +HD 98991 +2849 ± 3 +0.323 ± 0.004 +2455898 ± 5 +5.0 ± 0.3 +27.8 ± 0.8 +129.0 ± 1.3 +124.9 ± 0.7 +0.57 ± 0.08 +14.3 +5779 +HD 205011 +2846 ± 5 +0.23 ± 0.02 +2455297 ± 36 +5.3 ± 0.9 +34 ± 5 +56 ± 3 +74 ± 5 +0.61 ± 0.19 +1.36 +14392 +HD 204075 +2367 ± 9 +0.26 ± 0.05 +2455474 ± 109 +6.0 ± 0.4 +255 ± 15 +9 ± 165 +133 ± 5 +0.67 ± 0.10 +5.62 +41.1 +HD 20394 +2248 ± 8 +0.16 ± 0.06 +2456928 ± 105 +4.6 ± 0.5 +145 ± 18 +92 ± 6 +31 ± 3 +0.49 ± 0.10 +0.57 +489 +∗For the triple system HD 218356, we only include the outer orbit that hosts the WD. See Table 3 for the remaining parameters. +Article number, page 6 of 49 + +A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Table 2 continues +Star ID +P [days] +e +T0 [HJD] +a [AU] +ω∗ [◦] +Ω [◦] +i [◦] +MWD +Fit χ2 +χ2 +HGCA +HD 16458 +2017 ± 15 +0.098 ± 0.025 +2456430 ± 112 +4.2 ± 0.5 +119 ± 14 +109 ± 10 +61 ± 12 +0.74 ± 0.15 +2.53 +3888 +HD 5424 +1906 ± 17 +0.19 ± 0.05 +2455702 ± 145 +3.7 ± 0.4 +102 ± 14 +40 ± 74 +30 ± 3 +0.52 ± 0.11 +0.87 +445 +HD 49641 +1793 ± 21 +0.06 ± 0.06 +2456298 ± 352 +4.9 ± 0.9 +207 ± 63 +138 ± 126 +159.5 ± 1.6 +1.2 ± 0.4 +0.55 +1384 +HD 91208 +1770 ± 3 +0.178 ± 0.018 +2456240 ± 55 +4.2 ± 0.4 +83 ± 10 +167 ± 3 +133.6 ± 2.3 +0.83 ± 0.14 +6.19 +486 +HD 200063 +1743 ± 8 +0.07 ± 0.04 +2456510 ± 120 +4.3 ± 0.5 +228 ± 27 +152 ± 17 +115 ± 5 +0.95 ± 0.26 +0.96 +4910 +HD 201657 +1702 ± 4 +0.16 ± 0.7 +2456291 ± 280 +3.9 ± 0.6 +255 ± 61 +82 ± 4 +153 ± 3 +0.66 ± 0.18 +0.59 +1282 +HD 43389 +1688.4 ± 1.4 +0.083 ± 0.017 +2455663 ± 57 +4.0 ± 0.7 +190 ± 13 +168 ± 150 +111 ± 30 +0.76 ± 0.25 +1.03 +279 +HD 27271 +1681.1 ± 1.0 +0.224 ± 0.007 +2455519 ± 15 +4.2 ± 0.3 +210 ± 4 +6 ± 2 +103 ± 5 +0.70 ± 0.09 +3.93 +1441 +HD 95193 +1652 ± 5 +0.13 ± 0.02 +2456007 ± 52 +4.12 ± 0.15 +285 ± 10 +88 ± 37 +81 ± 25 +0.71 ± 0.08 +5.33 +268 +HD 210946 +1521.0 ± 0.9 +0.109 ± 0.011 +2455737 ± 35 +3.9 ± 0.5 +193 ± 8 +15 ± 9 +114 ± 8 +0.86 ± 0.26 +0.32 +231 +HD 127392 +1506.8 ± 1.8 +0.088 ± 0.017 +2456222 ± 91 +3.1 ± 0.6 +159 ± 20 +4 ± 175 +119 ± 15 +0.73 ± 0.26 +4.31 +1531 +HD 143899 +1461.8 ± 1.2 +0.18 ± 0.04 +2456481 ± 25 +3.66 ± 0.15 +279 ± 8 +88 ± 3 +125 ± 3 +0.66 ± 0.06 +2.00 +640 +HD 88562 +1451.0 ± 1.0 +0.203 ± 0.013 +2455931 ± 25 +2.9 ± 0.3 +352 ± 8 +61 ± 97 +87 ± 9 +0.48 ± 0.09 +0.84 +11.0 +HD 221531 +1402.3 ± 1.2 +0.165 ± 0.005 +2455563 ± 12 +3.0 ± 0.2 +189 ± 3 +136 ± 3 +59 ± 3 +0.58 ± 0.09 +0.49 +794 +HD 202400 +1391 ± 6 +0.25 ± 0.09 +2455555 ± 96 +2.9 ± 0.2 +37 ± 295 +86 ± 86 +61 ± 67 +0.65 ± 0.13 +1.48 +564 +HD 107574 +1384.8 ± 1.5 +0.083 ± 0.005 +2456521 ± 17 +2.99 ± 0.12 +216 ± 4 +22 ± 2 +166.3 ± 0.4 +0.74 ± 0.06 +0.24 +3766 +HD 58121 +1217 ± 5 +0.135 ± 0.019 +2455339 ± 50 +3.4 ± 0.5 +89 ± 10 +58 ± 28 +121 ± 4 +0.67 ± 0.19 +3.11 +149 +HD 31487 +1063.8 ± 0.4 +0.037 ± 0.018 +2455808 ± 98 +3.3 ± 0.2 +237 ± 33 +48 ± 2 +32.1 ± 1.2 +1.59 ± 0.22 +0.28 +1064 +HD 211594 +1018.5 ± 0.5 +0.058 ± 0.013 +2455675 ± 36 +2.8 ± 0.4 +76 ± 13 +65 ± 17 +123 ± 16 +0.55 ± 0.16 +1.85 +5.02 +HD 34654 +976.0 ± 0.3 +0.1114 ± 0.0016 +2455212 ± 2 +2.36 ± 0.12 +326.5 ± 0.7 +91 ± 4 +74 ± 5 +0.64 ± 0.06 +0.058 +183 +HD 92626 +921.7 ± 1.7 +0.014 ± 0.014 +2455685 ± 237 +3.0 ± 0.5 +116 ± 199 +92 ± 62 +85 ± 9 +0.90 ± 0.27 +0.61 +2.47 +HD 50264 +910.2 ± 0.8 +0.077 ± 0.018 +2455933 ± 43 +2.1 ± 0.2 +238 ± 17 +63 ± 17 +103 ± 40 +0.63 ± 0.13 +0.53 +246 +HD 49841 +897.5 ± 1.9 +0.162 ± 0.015 +2455518 ± 21 +2.80 ± 0.10 +350 ± 6 +110 ± 51 +109 ± 19 +0.82 ± 0.16 +0.51 +14.5 +HD 58368 +673.1 ± 1.5 +0.22 ± 0.02 +2455715 ± 28 +2.23 ± 0.17 +18 ± 6 +99 ± 68 +78 ± 27 +0.66 ± 0.17 +0.23 +0.98 +HD 207585 +671.7 ± 0.4 +0.03 ± 0.03 +2455613 ± 174 +1.72 ± 0.16 +109 ± 207 +103 ± 17 +93 ± 20 +0.57 ± 0.12 +1.06 +20.6 +HD 44896 +629.0 ± 1.1 +0.019 ± 0.013 +2455388 ± 76 +2.29 ± 0.10 +228 ± 35 +101 ± 35 +78 ± 35 +1.0 ± 0.2 +0.99 +5.86 +HD 199939 +585.39 ± 0.09 +0.281 ± 0.012 +2455207 ± 4 +2.12 ± 0.18 +48 ± 3 +110 ± 84 +82 ± 21 +0.73 ± 0.14 +2.24 +4.61 +HD 123585 +460.1 ± 1.4 +0.03 ± 0.04 +2455534 ± 152 +1.41 ± 0.11 +191 ± 73 +27 ± 135 +90 ± 19 +0.65 ± 0.11 +1.53 +7.32 +HD 24035 +378.0 ± 0.5 +0.014 ± 0.016 +2455307 ± 184 +1.41 ± 0.15 +204 ± 110 +52 ± 106 +71 ± 31 +0.76 ± 0.25 +0.74 +6.69 +HD 224621 +308.20 ± 0.11 +0.020 ± 0.012 +2455264 ± 32 +1.03 ± 0.06 +309 ± 57 +64 ± 27 +31 ± 7 +0.66 ± 0.26 +3.05 +59.7 +HD 87080 +274.31 ± 0.05 +0.162 ± 0.016 +2455230 ± 4 +1.05 ± 0.10 +128 ± 5 +149 ± 112 +60 ± 12 +0.70 ± 0.18 +3.58 +15.6 +HD 121447 +185.65 ± 0.05 +0.012 ± 0.010 +2455243 ± 51 +0.85 ± 0.05 +267 ± 104 +90 ± 50 +59 ± 76 +0.72 ± 0.36 +4.05 +8.71 +HD 77247 +80.5371 ± 0.0013 +0.108 ± 0.005 +2455236.5 ± 0.7 +0.63 ± 0.03 +40 ± 3 +60 ± 35 +98 ± 25 +0.54 ± 0.11 +3.11 +2.31 +Article number, page 7 of 49 + +A&A proofs: manuscript no. main +Fig. 1: Orvara results for the main-sequence Ba star HD 2454. Top: astrometric and spectroscopic orbits. The RV plot includes +radial-velocity measurements from CORAVEL (orange circles), SOPHIE (pink diamonds), and HERMES (green triangles). Bot- +tom: Hipparcos and Gaia proper motions. In all plots, the best-fitting orbit is plotted as a black thick line, while 40 other well-fitting +orbits are included and colour-coded as a function of the companion mass. +the parameters of both orbits in Appendix B. Only the outer orbit +information is listed together with the other WD orbits in Table +2. +5.1.2. HD 201657 +Our orbit fit for HD 201657 converged to twice the published +orbital period and to a much more eccentric orbit. The astromet- +ric data favours the longer orbit, and the RV data is not very +constraining since we have only 15 CORAVEL points and one +HERMES point. However, given the eccentricity-period diagram +of Ba stars, the orbit published by Jorissen et al. (2019), the least +eccentric of the two, is the most likely. We attempted to recover +this orbit in order to check the quality of such a fit and calcu- +late the WD companion mass by including an orbital eccentricity +prior of 0.15±0.15. We recovered Jorissen et al. (2019)’s orbital +solution, although with a slightly higher χ2 for the astrometric +data. Since we considered this solution more likely for a Ba star, +we listed this orbit in Table 2, but we show both fits and corner +plots in Appendix C. More HERMES data would be essential to +solve this case. +5.2. Astrometric orbital parameters +Finally, in addition to the new and improved orbital parameters, +this method provided us with orbital inclinations for all these Ba +star systems. Fig. 4 shows the distribution of the obtained cos(i) +values. This distribution should be flat if we could assume our +sample of binaries is randomly distributed on the sky, and even +though we only have 60 systems, the distribution is compatible +with a uniform one. We performed a Kolmogorov-Smirnov (KS) +test (e.g. Press et al. 1986), and we obtained p-values higher than +0.8 when comparing our cos(i) distribution with uniform distri- +butions of the same sample size. +The new orbital parameters are also compatible with the as- +trometric parameters published in Gaia DR3 for the ten targets +available in their catalogue. Concerning the periods, all Gaia +DR3 values are consistent with our values within 2σ. The largest +difference is found for HD 221531, for which Gaia DR3 pub- +lished a period of 1668 ± 135 days, about 260 days longer than +our period. The Gaia DR3 time span is about 1000 days (Gaia +Collaboration et al. 2022), while our data covers a few decades +in most cases. Hence, we think that our method is more reliable +to obtain the periods of long-period binaries. The eccentricities +Article number, page 8 of 49 + +3000 +CORAVEL +HERMES +SOPHIE +2000 +Mcomp(Mo) +RV (m/s) +1.25 +1000 +0.70 +1.00 +0.65 +0.75 +0 +0.60 +[arcsec] +0.50 +0.55 +0.25 +-1000 +0.50 +0.00 +0.45 +0.25 +1000 +大 +0.40 +2027 +O-C +-0.50 +0 +0.35 ++ +f +-0.75 +-1000 +0.30. +0.5 +0.0 +0.5 +-1.0 +1980 +1990 +2000 +2010 +2020 +20 +Aα (arcsec) +Epoch (yr) +HIPPARCOS +HIPPARCOS +65 +-180 +GAIA +GAIA +60 +-185 +55 +(mas/yr) +/yr) +-190 +(mas/ +50 +*n +45 +195 +40 +-200 +35 +-205 +30 L +0.5 +1 +O-C +0-0 +0 +0.0 +-0.5 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (year) +Epoch (year)A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Mpri (M ) = 1.24+0.30 +0.30 +0.30 +0.45 +0.60 +0.75 +0.90 +Msec (M ) +Msec (M ) = 0.503+0.092 +0.086 +20 +30 +40 +50 +60 +a (AU) +a (AU) = 22.2+4.3 +2.9 +0.48 +0.56 +0.64 +0.72 +e +e = 0.588+0.040 +0.042 +0.4 +0.8 +1.2 +1.6 +2.0 +Mpri (M ) +32 +40 +48 +56 +i ( ) +0.30 +0.45 +0.60 +0.75 +0.90 +Msec (M ) +20 +30 +40 +50 +60 +a (AU) +0.48 +0.56 +0.64 +0.72 +e +32 +40 +48 +56 +i ( ) +i ( ) = 33.9+5.8 +3.2 +Fig. 2: Corner plot of some derived parameters for HD 2454 including mass of the two stars, the semimajor axis, the eccentricity, +and orbital inclination. +are compatible as well, without significant exceptions, and fi- +nally, we used the Thiele-Innes elements published in the Gaia +DR3 catalogue and followed Halbwachs et al. (2022) to compute +the orbital inclinations of these systems from the Gaia DR3 data. +The Gaia DR3 inclinations are also compatible with the inclina- +tions we obtained with our full RV+astrometric model within +1.2 times our σ. As discussed above, the HGCA is not very con- +straining for systems with periods below about 3 years, so while +we think our method is better to determine the orbital periods of +Ba stars, the Gaia DR3 inclinations are probably of better qual- +ity than ours for the shorter-period systems. When the epoch as- +trometry of the Gaia mission is published, we will be able to +combine these data with all our other data sets and improve our +results for the shortest period systems. +5.3. White Dwarf masses +Table 2 lists the masses we obtained for the companions to all the +Ba stars in our sample, and Fig. 5 shows the distribution of these +masses as a purple dashed histogram. Also in Fig. 5 we compare +this new distribution to the distribution obtained by Jorissen et al. +(2019) and Escorza et al. (2019a) for the same stars, which is +drawn in black. The insert in the figure shows the cumulative +frequency of the two distributions, including an envelope with +the 1 − σ uncertainty for our distribution, which also envelopes +the old distribution. We obtained a p-value of 0.010 on a KS test, +Table 3: Orbital parameter of the triple system HD 218356 +Parameter +Inner orbit +Outer orbit +Period, P [days] +111.15+0.03 +−0.03 +15194+2600 +−1600 +Eccentricity, e +0.072+0.048 +−0.045 +0.39+0.13 +−0.12 +T. of periastron, T0 [HJD] +2455289+15 +−85 +2469014+2800 +−2800 +Semimajor axis, a [AU] +0.79+0.10 +−0.08 +22.1+3.6 +−2.8 +Arg. of periastron, ω [◦] +55+270 +−37 +73+21 +−24 +Ascending node, Ω [◦] +90+60 +−62 +153+14 +−17 +Inclination [◦] +90+42 +−41 +157+4 +−5 +Companion mass [M⊙] +0.13+0.06 +−0.03 +0.85+0.25 +−0.18 +which is not low enough to reject the null hypothesis. The two +distributions are not statistically different. +In Fig. 6, we plot the mass distributions of the companions to +strong and mild Ba giants, separately. As mentioned in the intro- +duction, this distinction is made based on the abundance ratios +[La/Fe] and [Ce/Fe] and on the Ba index introduced by Warner +(1965). We do not include the pre-RGB stars in this comparison, +because the distinction between strong and mild enhancement +Article number, page 9 of 49 + +.t.A&A proofs: manuscript no. main +Fig. 3: Best fitting models to the RV data of HD 218356 +Fig. 4: Distribution of cos i where i are the orbital inclinations +of the Ba star systems. Bin-width chosen to roughly follow the +Freedman–Diaconis rule (Freedman & Diaconis 1981). +has not been as clearly established as it has for the giants. We +note that the WDs occupying the high-mass tail belong to sys- +tems with strong Ba giants. However, we performed a KS test, +and we obtained a p-value of 0.45, meaning that we cannot re- +ject that the two samples are drawn from the same distribution. +The cumulative distributions plotted in the insert also show that +Fig. 5: Mass distribution of WD companions of Ba stars. The +purple histogram corresponds to the WD masses obtained for +this publication, while the black histogram includes the results +published in Jorissen et al. (2019) and Escorza et al. (2019a) as- +suming a narrow distribution of Q and MWD. The bin-width was +chosen to roughly follow the Freedman–Diaconis rule (Freed- +man & Diaconis 1981). The insert in the figure shows the cu- +mulative frequency of the same two samples, including a 1-σ +envelope for our results. +Fig. 6: Mass distribution of WD companions of strong (blue) +and mild (dashed green) Ba giants. The insert in the figure shows +the cumulative frequency of the same two samples, including a +1-σ envelope for our results. +taking the 1 − σ uncertainty into account, the distributions are +not very different. +There are a few individual systems that appeared as clear +outliers or that even have WDs with unphysical masses. These +are briefly discussed below. +Article number, page 10 of 49 + +0.6 +0.5 +Probability density +0.4 +0.3 +0.2 +0.1 +0.0 +-1.0 +-0.5 +0.0 +0.5 +1.0 +cos(i)4.0 +Jorissen19 & Escorza19 +This work +3.5 +1.0 +0.8 +Probability density +3.0 +0.6 +2.5 +0.4 +2.0 +0.2 +1.5 +0.250.500.751.00 +1.25 +1.501.75 +M/Mo +1.0 +0.5 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +M/moWD with strong Ba giants +3.0 +WD with mild Ba giants +1.0 +2.5 +0.8 +Probability density +0.6 +2.0 +0.4 +1.5 +0.2 +1.0 +0.0 +0.5 +1.0 +1.5 +M/Mo +0.5 +0.0 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +2.00 +M/MoCORAVEL +3000 +DAO +HERMES +1.4 +2000 +(s/w) +1000 +1.2 +RV +0 +1.0 +-1000 +-2000 +0.8 +-3000 +2500 +O-C +0.6 +0 +-2500 +1980 +1990 +2000 +2010 +2020 +Epoch (yr) +4000 +3000 +3000 +2000 +RV (m/s) +1000 +(m/s) +2000 +0 +RV +1000 +-1000 +0 +-2000 +-3000 +-1000 +2500 +2500 +O-C +O-C +0 +0 +2500 +-2500 +1982 +1984 +1986 +1988 +2002 +2003 +2004 +2005 +2006 +Epoch (yr) +Epoch (yr)A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +5.3.1. The least massive WDs: HD 18182 and HD 95241 +There are two systems for which our simulations converged to +very low WD masses. These are HD 18182 and HD 95241. The +fit we achieved for the former is less than ideal (see Figs. A.14 +and A.16), and even though the mass is small, taking the error +bars into account, the value is compatible with an average WD in +our sample. The CORAVEL RV data is not very constraining and +the HERMES points, being of much higher quality, still fall on +the same range of orbital phases, covering in total less than half +of the orbit. Additionally, the Hipparcos and Gaia proper mo- +tions in the right ascension direction are very similar, not adding +strong constraints to the fit either. This WD mass should be taken +with caution. +The fit for HD 95241, on the other hand, is significantly bet- +ter. We used 97 RV points that cover very well the whole orbit +(see Fig. A.21) and obtained clean and symmetric posterior dis- +tributions (see Fig. A.23). Of course, MBa and MWD are very +strongly correlated, so if the MBa prior was incorrect, too small +in this case, it would directly affect MWD. The mass of HD 95241 +was determined by Escorza et al. (2019b) by comparing the loca- +tion of the star on the HR diagram with STAREVOL (Siess et al. +2000; Siess & Arnould 2008) evolutionary tracks. The stellar pa- +rameters were determined from HERMES high-resolution high- +signal-to-noise spectra and are in agreement with other studies +(e.g. Takeda 2007; Soubiran et al. 2016). However, HD 95241 +was flagged as a mild Ba dwarf by Edvardsson et al. (1993) hav- +ing only a marginal overabundance of s-process elements with +respect to iron. Other Ba dwarf candidates of their sample have +been proven to be wrongly flagged. Most of them are likely sin- +gle stars (Escorza et al. 2019b). It is possible that HD 95241 has +a low-mass companion that is not a WD, and if it is a WD, its +AGB progenitor was not massive enough to reach the thermally +pulsing AGB phase and produce s-process elements. HD 95241 +is likely not a Ba star and will be removed from further analysis. +5.3.2. The most massive WDs: HD 49641 and HD 31487 +On the high-mass end of the distributions, there are two systems +with WD masses clearly outlying from the initial mass distri- +bution (MWD ≥ 1.2 M⊙ ). These are HD 49641, with MWD = +1.2 ± 0.4 M⊙ , and HD 31487, with MWD = 1.59 ± 0.22 M⊙ . +The fit for HD 49641 is not very good, because the available RV +data was scarce and old, so one should take this WD mass with +caution, but the fits for HD 31487 seems reliable, including a +clean result for the orbital projection on the sky (see Fig. 7). In +order to try to explain this last mass, one could again try to in- +voke a wrong MBa prior. We used the primary mass determined +by Karinkuzhi et al. (2018). The primary mass listed by Jorissen +et al. (2019) is not in agreement with Karinkuzhi et al. (2018)’s +within error bars, but we decided to use the latter after study- +ing their HR diagram (their Fig. 16). In any case, Jorissen et al. +(2019)’s mass is higher, and would result in a higher companion +mass. Karinkuzhi et al. (2018)’s value seems reasonable given +the location of the star on the HR diagram, and it is a very aver- +age value for giant Ba stars. Additionally, there is no big discrep- +ancy between the parallaxes published in the different Gaia Data +Releases. While a wrong parallax could have led to a wrong lu- +minosity, hence mass, determination, we have no good reason to +doubt this mass. From the posterior distributions and 1D projec- +tions shown in Fig. A.87, one can see that a significantly lower +MBa could lower MWD within the Chandrasekhar limit (about +1.4 M⊙ ; Chandrasekhar 1939), but that the dynamics of this sys- +tem do not favour a secondary mass below ∼1.2 M⊙ . +0.01 +0.00 +0.01 + (arcsec) +0.015 +0.010 +0.005 +0.000 +0.005 +0.010 +0.015 + [arcsec] + 2022 +Astrometric Orbits +1.2 +1.4 +1.6 +1.8 +2.0 +Mcomp(M ) +Fig. 7: Projection on the sky of HD 31487. +Only with the dynamical information that we currently have, +it is difficult to confirm that this ’massive companion’ is a single +object, and not a close pair formed, for example, by a faint main- +sequence star and a WD (see van den Heuvel & Tauris 2020 for +an example of such a situation). The strong s-process enhance- +ment strongly suggests that there is a WD in the system, but +since we cannot be certain of its mass, HD 31487 will be re- +moved from further discussion. +6. Discussion +6.1. Mass distributions +The mass distribution that we obtained for the WD companions +of Ba stars is compatible with current estimates for field WD +masses. The average mass of DA WDs (WDs with only Balmer +lines in their spectra) is about 0.60 M⊙ , while that of DB WDs +(WDs with no H or metals lines in their spectra, only helium +lines) is 0.68 M⊙ (Kleinman et al. 2013). The weighted average +of our mass distribution is 0.65 M⊙ , after removing the two tar- +gets mentioned in Sect. 5.3. There is a high-mass tail present +in the mass distribution of WDs orbiting Ba giant that Jorissen +et al. (2019) and Escorza et al. (2019a) already discussed (see +also Fig. 5). +In order to evaluate if Q = M3 +WD/(MBa + MWD)2 is constant, +we computed this value for all our targets and present the average +and the standard deviation for each one of the three subsamples +separately in Table 4. The new distributions are marginally dif- +ferent to literature Q distributions (see Table 1 in Escorza et al. +2019a). We obtained p = 0.048 for the strong Ba giants, p = +0.035 for the mild Ba giants and p = 0.012 for the Ba dwarfs, +when we performed KS tests. The main difference is that the new +distributions are not as narrow as obtained in the past when mod- +elling f(m) = Q sin3 i, with f(m) being the spectroscopic mass +function. In order to check if this is caused by the fact that the in- +dividual inclination uncertainties play a role now, while an incli- +nation distribution was assumed in the past, we calculated new +Q distributions removing the 10 and 20% systems with larger +uncertainties. All the observed distributions are broader than the +literature ones, but not significantly different. +In Table 4, we have also included the average and standard +deviations of the current mass ratios of the three Ba star subsam- +ples. The two subsamples of giants show closer values, with the +Article number, page 11 of 49 + +A&A proofs: manuscript no. main +Table 4: Average and standard deviations of the Q-values, with +Q = M3 +WD/(MBa + MWD)2, and the mass ratios of strong Ba gi- +ants, mild Ba giants and pre-RGB Ba stars. +Ba type +Q-value +Mass ratio (q) +Strong Ba giant +0.054 ± 0.022 +0.37 ± 0.09 +Mild Ba giant +0.036 ± 0.019 +0.28 ± 0.08 +Ba dwarf +0.091 ± 0.035 +0.60 ± 0.14 +average mass ratio of strong Ba giants being slightly higher than +that of mild Ba giants, in agreement with what Jorissen et al. +(2019) reported. We perform a KS test and obtained a p-value of +0.015, which is not low enough to statistically confirm that this +difference. The average mass ratio of Ba dwarfs is much higher, +but the currently known Ba dwarfs are significantly less massive +than the giants (Escorza et al. 2019b), then accounting for this +result. +6.2. A comment on nucleosynthesis predictions +It is difficult to make a direct correlation between the WD com- +panion mass and the s-process enhancement of the Ba star be- +cause many parameters, apart from the WD progenitor mass, +strongly affect the final Ba star abundances and the unknowns are +still stronger than the observational constraints (see Cseh et al. +2022, for a study of abundances in individual Ba giant systems). +For example, the efficiency of the mass transfer and the dilution +factor, the ratio between the accreted mass and the mass in the Ba +star envelope over which this is mixed in, are major uncertain- +ties in our understanding of the formation of Ba stars and will +directly affect the final s-process enhancement (e.g. Stancliffe +2021). Of course, the efficiency of the s-process of nucleosynthe- +sis in the interiors of AGB stars, which strongly depends on the +mass and the metallicity of the star itself, is also a key parameter +in order to explain a possible correlation between WD mass and +Ba enhancement(e.g. Busso et al. 2001; Karakas & Lugaro 2016; +Van der Swaelmen et al. 2017). Additionally, even the number of +thermal pulses and third dredge-ups experienced by the AGB star +before the mass transfer episode took place will have an effect on +the final s-process abundance pattern (e.g. Shetye et al. 2018), as +well as mixing and diffusion below the AGB star’s convective +envelope will (e.g. Goriely & Siess 2018). +Standard stellar-evolution models do not predict solar- +metallicity low-mass AGB stars undergo third dredge-ups (e.g. +Cristallo et al. 2015; Karakas & Lugaro 2016). This limit can go +down to 1 M⊙ at lower metallicities (e.g. Stancliffe et al. 2005; +Lugaro et al. 2012; Fishlock et al. 2014). However, including +different additional effects in the models can help, for exam- +ple, Weiss & Ferguson (2009) showed that including some over- +shooting below the convective pulse, their models could make a +1 M⊙ AGB star undergo third dredge-ups. Additionally, Shetye +et al. (2019, 2021) found several low-mass AGB stars currently +undergoing third dredge-ups and their models succeeded to re- +produce the s-process overabundance including diffusive mixing +at the bottom of the stellar envelope. Additionally, according to +several studies, the AGB stars that polluted Ba stars need to have +masses below 3 M⊙ to be able to reproduce their abundance ra- +tios with models (Lugaro et al. 2003a, 2012, 2016; Cseh et al. +2018; Karinkuzhi et al. 2018). +Figure 8 shows the relation between the metallicity (listed in +Table 1) and the obtained WD masses (Table 2) for the preRGB +Ba stars (orange circles), the strong Ba giants (blue squares) +and the mild Ba giants (green triangles) in our sample. The fig- +ure shows an expected correlation between the Ba-type and the +metallicity, caused by the fact that the efficiency of the s-process +in AGB stars decreases as the metallicity increases (e.g. Cseh +et al. 2018; Jorissen et al. 2019). However, there is no obvi- +ous correlation between the WD mass and the metallicity, even +though the AGB mass directly affects the s-process efficiency as +well. The least massive WDs are in systems with [Fe/H] < −0.1, +in agreement with the models, and the most massive WDs ac- +company Ba giants of [Fe/H] between −0.4 and −0.2, with the +three most massive WDs being in a strong Ba star systems. +Among our sample of 58 systems (after having removed +(HD 95241 and HD 31487 from the WD sample), we do not find +Ba stars with unexpectedly low mass companions. As discussed +in Sect. 5.3, the companion mass for HD 18182 should be taken +with caution, but all other Ba star systems have WDs of around +or more massive than 0.5 M⊙ , meaning that their progenitors +were AGB stars of around or more massive than 1 M⊙ . Note that +to make such a statement, one needs to rely on initial-final mass +relationships (IFMR). We used as a reference the relation pub- +lished by Marigo et al. (2020, 2022). Using the same relation, we +can claim that a fraction of the AGB stars that polluted our sam- +ple of Ba stars were more massive than the expected 3 M⊙ limit, +since we found that several WDs have masses around or higher +than 0.8 M⊙ . This is the case even taking into account the kink +that Marigo et al. (2020, 2022) find for WDs of about 0.70 – +0.75 M⊙ with carbon AGB progenitors. Most IFMRs (e.g. Wei- +demann 2000; Kalirai et al. 2008; Williams et al. 2009; Andrews +et al. 2015; Cummings et al. 2016; El-Badry et al. 2018) flatten at +around MWD ∼ 0.8 M⊙ , making stars with a wide range of initial +masses accumulate at that WD mass. However, their progenitors +are expected to have initial masses in the range between 3.5 and +5.5 M⊙ , hence more massive than what the Ba stars abundance +ratios seem to indicate. +The presence of these massive WDs orbiting around both +strongly and mildly polluted Ba stars presents important con- +straints, as well as an interesting challenge, for evolutionary and +nucleosynthesis models. Future studies of these systems follow- +ing the line presented by Stancliffe (2021) or Cseh et al. (2022), +but using these new WD masses, might be able to tell us new +things about AGB stars. We note that our error bars are signif- +icant and that these statements blur if we consider two or three +sigma uncertainties. This will improve when we have NSS paral- +laxes to obtain more accurate Ba star masses and Gaia astromet- +ric epochs to improve the RV+astrometry fit. Direct imaging ob- +servations could also help constrain the longest-period systems +better (see Sect. 7). +7. Future observational prospects: direct imaging +of these white dwarfs with SPHERE +The nearby (d ≲ 100 pc) Ba stars that are host to long-period +(P ≳ 10 yr) companions are suitable candidates for high-contrast +imaging observations to spatially resolve the companion. These +observations would provide relative astrometric and photometric +measurements between the WD and the Ba star host. A single +measurement of the instantaneous angular separation between +the components would constrain both the total semi-major axis, +and thus the total system mass, and the inclination (unless the +observation occurred when the companion was crossing the line +of nodes). Photometric measurements of the companion could +be used to estimate the bolometric luminosity of the compan- +ion which, in conjunction with the mass, can be compared to +Article number, page 12 of 49 + +A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. 8: Relation between the metallicity (Table 1) and the ob- +tained WD masses (Table 2) for the preRGB Ba stars (orange +circles), the strong Ba giants (blue squares) and the mild Ba gi- +ants (green triangles) in our sample. +WD cooling models to estimate the age of the companion (see +e.g. Bonavita et al. 2020 for the discovery and analysis of a WD +companion around a K-type star with SPHERE and e.g. Grat- +ton et al. 2021 for the study of a sample of Sirius-like systems, +long-period main-sequence + white dwarf binaries). +We assessed the feasibility of spatially resolving the compan- +ion by comparing the predicted angular separation and flux ratio +between the WD companion and the Ba host star to the perfor- +mance of the VLT/SPHERE instrument (Beuzit et al. 2019). We +filtered the sample to exclude systems with a median apoastron +distance within 100 mas, calculated from the MCMC samples +described in Section 4. For these systems, the companion would +always be within the inner working angle of the instrument, and +impossible to resolve with SPHERE. This filter resulted in a sub- +sample of eight systems for which the companion will be at a +projected separation of ρ > 100 mas at some point in its orbit. +The feasibility of direct detection also depends on the flux +ratio between the WD companion and Ba star host. We esti- +mate the H-band flux ratio for each MCMC sample using pure- +hydrogen (DA) atmosphere mass-luminosity relations from Hol- +berg & Bergeron (2006). We assigned an age to each MCMC +sample at random from a uniform distribution between 106 and +1010 yr to account for the unknown age of the WD. The model +grid was linearly interpolated in (log t, M) to extract an absolute +H-band magnitude. This was converted into a flux ratio using the +parallax from the MCMC sample and the apparent H-band mag- +nitude of the Ba star. We assumed the companion has negligible +flux in the H-band relative to the Ba star, such that the catalogue +H-band magnitude of the system can be entirely ascribed to the +Ba star. +We converted the orbital elements to the angular separation +between the WD and the Ba star host at the epochs 2023, 2024, +and 2025 for each MCMC sample. The predicted angular sepa- +ration and flux ratio for each sample was then compared to the +SPHERE contrast curve given in Wahhaj et al. (2021). We ac- +counted for the degradation in contrast performance for fainter +stars (e.g. Jones et al. 2022) by scaling the contrast curve by the +square root of the H-band flux ratio between the Ba star host and +HR 8799, the star observed by Wahhaj et al. (2021) to measure +the contrast curve. +The predicted separations in 2023 and H-band contrast for +each of the eight systems are shown in Figure 9. There are six +systems with a non-negligible probability of detection at this +epoch; the others are too faint to be detected given the expected +contrast curve. The majority of these systems exhibit a strong +correlation between the separation at the 2023 epoch and the +mass of the WD companion. This can partly be explained by the +constraint provided by a direct measurement of the semi-major +axis of the system, leading to a much more precise measurement +of the total mass of the system. +8. Summary and conclusions +The WD companions of Ba stars contain important information +about the formation of these chemically peculiar stars, about the +binary interaction processes that these systems underwent in the +past, and about the nucleosynthesis processes that took place in- +side their AGB progenitors. However, they are cool, dim, and +generally not detected by direct methods, so they have not been +studied in detail in the past. A few absolute masses had been +determined before this work by combining the spectroscopic or- +bital parameters of these systems with Hipparcos astrometric +data. However, most published masses for WD companions of +Ba stars were computed by making assumptions on the relation +between the masses of the two stellar components in these sys- +tems or on their orbital inclinations. +In this work, we used the software package orvara to com- +bine radial-velocity data, Hipparcos and Gaia positions and +proper motions through the Hipparcos-Gaia Catalogue of Ac- +celerations, and astrometric epoch measurements from the Hip- +parcos mission, and determine the astrometric orbital parameters +of 60 stars flagged as Ba dwarfs or giants. Using this method, we +could constrain the orbits of two long-period systems that could +not have been constrained before with RV data only, and we im- +proved the orbital solution of a few other systems. Orbital incli- +nations were also determined for the first time for many of these +systems, and finally, including a prior on the Ba star masses, we +also derived the mass of the secondary stars in these systems. +Finally, we discovered that HD 218356, one of the shortest pe- +riod Ba star systems known, is actually a triple system. We de- +termined the parameters of both the inner and outer orbits and +the masses of the two components, and it is very likely that the +WD companion that polluted HD 218356 is in the outer orbit, +explaining the mild s-process enhancement of the Ba giant. +The WD mass distribution presented in this work includes +all systems published by Jorissen et al. (2019), Escorza et al. +(2019b) and North et al. (2020) that had a single-star Hipparcos +solution and that were not confirmed triples. This mass distri- +bution is compatible with field WD mass distributions and with +those published before for Ba stars. The distribution extends to +high WD masses, higher than expected by theoretical models of +the s-process of nucleosynthesis that have focused on reproduc- +ing the abundance ratios measured on Ba star atmospheres. This +work brings new observational constraints for these models and +an interesting challenge to our understanding of the formation of +Ba stars. +In order to look at Ba stars with new eyes, we plan future +direct imaging observations of six of the longest-period systems +with SPHERE. On the one hand, this data will provide us with +a measurement of the instantaneous angular separation between +the components of the system, partially breaking the total mass +- semimajor axis correlation and helping us get more accurate +Article number, page 13 of 49 + +preRGB Ba stars +strong Ba giants +1.50 +mild Ba giants +1.25 +Table +2 +1.00 +MwD/Mo, +0.75 +0.50 +0.25 +0.00 +0.8 +0.6 +0.4 +0.2 +0.0 +0.2 +[Fe/H], Table 1A&A proofs: manuscript no. main +8 +10 +12 +14 +16 +BD-11 3853 +HD2454 +HD95241 +HD98991 +0.0 +0.1 +0.2 +0.3 +0.4 +ρ at 2023 (asec) +8 +10 +12 +14 +16 +∆H (mag) +HD104979 +0.0 +0.1 +0.2 +0.3 +0.4 +HD139195 +0.0 +0.1 +0.2 +0.3 +0.4 +HD182274 +0.0 +0.1 +0.2 +0.3 +0.4 +HD218356 +Fig. 9: Predicted angular separation in 2023 and contrast of the eight systems with median apoastron distances of > 100 mas. +Contours indicate 1, 2, and 3σ credible regions. The predicted SPHERE contrast is given by the solid red line, and the red shaded +region corresponds to the inner working angle of the instrument. Six of the systems are amenable to direct detection in the near +future. +masses. On the other hand, we will be able to estimate the +bolometric luminosity of the WD, which combined with its +mass, can be compared to WD cooling models to estimate the +age of these systems. +Acknowledgements. The authors thank Prof. Dr. Alain Jorissen and Dr. Henri +Boffin for the enriching discussions and Prof. Dr. Hans Van Winckel for pro- +viding us with a few new, unpublished HERMES RV points for our targets. +We also want to thank the referee, Dr. Carine Babusiaux, for helping us to im- +prove this manuscript. We are grateful to all observers of the HERMES con- +sortium for their time dedicated to the Mercator-HERMES long-term binary- +monitoring program. The HERMES spectrograph is supported by the Fund +for Scientific Research of Flanders (FWO), Belgium, the Research Council +of KU Leuven, Belgium, the Fonds National de la Recherche Scientifique +(F.R.S.-FNRS), Belgium, the Royal Observatory of Belgium, the Observa- +toire de Genève, Switzerland and the Thüringer Landessternwarte Tautenburg, +Germany. This publication includes data retrieved from the SOPHIE and the +ELODIE archives at Observatoire de Haute-Provence (OHP), available at http: +//atlas.obs-hp.fr/sophie and http://atlas.obs-hp.fr/elodie, re- +spectively. This work makes use of the "Synthetic Colors and Evolutionary +Sequences of Hydrogen- and Helium-Atmosphere White Dwarfs" hosted at +http://www.astro.umontreal.ca/~bergeron/CoolingModels and of the +SIMBAD database, operated at CDS, Strasbourg, France. 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A., Bolte, M., & Koester, D. 2009, ApJ, 693, 355 +Article number, page 15 of 49 + +A&A proofs: manuscript no. main +Appendix A: RV curves, proper motions and corner +plots +Article number, page 16 of 49 + +A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. A.1: RV curve and proper motions of HD 2454 +Fig. A.2: RV curve and proper motions of HD 119185. We used a fixed RV offset of 500 m/s (Jorissen et al. 2019). +Mpri (M ) = 1.24+0.30 +0.30 +0.30 +0.45 +0.60 +0.75 +0.90 +Msec (M ) +Msec (M ) = 0.503+0.092 +0.086 +20 +30 +40 +50 +60 +a (AU) +a (AU) = 22.2+4.3 +2.9 +0.48 +0.56 +0.64 +0.72 +e +e = 0.588+0.040 +0.042 +0.4 +0.8 +1.2 +1.6 +2.0 +Mpri (M ) +32 +40 +48 +56 +i ( ) +0.30 +0.45 +0.60 +0.75 +0.90 +Msec (M ) +20 +30 +40 +50 +60 +a (AU) +0.48 +0.56 +0.64 +0.72 +e +32 +40 +48 +56 +i ( ) +i ( ) = 33.9+5.8 +3.2 +Fig. A.3: Corner plot of HD 2454 +Mpri (M ) = 1.70+0.20 +0.20 +0.6 +0.8 +1.0 +1.2 +Msec (M ) +Msec (M ) = 0.651+0.079 +0.060 +24 +32 +40 +a (AU) +a (AU) = 22.5+3.0 +2.0 +0.5 +0.6 +0.7 +0.8 +0.9 +e +e = 0.611+0.079 +0.070 +1.2 +1.5 +1.8 +2.1 +Mpri (M ) +75 +90 +105 +120 +i ( ) +0.6 +0.8 +1.0 +1.2 +Msec (M ) +24 +32 +40 +a (AU) +0.5 +0.6 +0.7 +0.8 +0.9 +e +75 +90 +105 +120 +i ( ) +i ( ) = 98+10 +13 +Fig. A.4: Corner plot of HD 119185 +Article number, page 17 of 49 + +3000H +CORAVEL +HIPPARCOS +HIPPARCOS +65 E +至 +-180 +SOPHIE +GAIA +GAIA +HERMES +60 E +0.70 +2000 +-185 +55 E +0.65 +(mas/yr) +(uk/sew) +RV (m/s) +-190 +1000 +50E +0.60 +*0r +-195 +45 E +oF +40E +0.50 +-200 +-1000 +0.45 +35 +-205 +0.40 +30 +1E +0.5 +1000 +0.35 +O-C ++T +0 +0.0 +0 +-1000 +0.30 +-11 +-0.5E +1990 +2000 +2010 +2020 +1990 +1995 +20002005 +2010 + 2015 +19901995 +2000 +2005 +20102015 +Epoch (yr) +Epoch (year) +Epoch (year)4000 +CORAVEL +■HIPPARCOS +-28 +IHERMES + GAIA +21.5 +30000 +0.90 +-29 [ +2000 +21.0 +0.85 +(mas/yr) +(mas/yr) +1000F +RV (m/s) +-30 +20.5 +0.80 +20.0 +0.75 +-31 +-1000E +19.5 +0.70 +-2000E +-32 +0.65 +-3000 +19.0 +HIPPARCOS +-33 + GAIA +-4000 +0.60 +1000斤 +2.5 +0.5 +O-C +O-C +0.55 +0.0 +0.0 +-2.5 日 +-0.5 +-1000 +1990 +2010 +19901995 +2000 +2005 +20102015 +1990 +2000 +2020 +1995 +2000 +2005 + 2010 + 2015 +Epoch (yr) +Epoch (year) +Epoch (year).t.A&A proofs: manuscript no. main +Fig. A.5: RV curve and proper motions of BD-11o3853 +Fig. A.6: RV curve and proper motions of HD 104979 +Mpri (M ) = 0.85+0.15 +0.15 +0.50 +0.75 +1.00 +1.25 +Msec (M ) +Msec (M ) = 0.76+0.14 +0.10 +25 +50 +75 +100 +125 +a (AU) +a (AU) = 18.6+5.6 +2.7 +0.45 +0.60 +0.75 +e +e = 0.460+0.052 +0.035 +0.4 +0.6 +0.8 +1.0 +1.2 +Mpri (M ) +75 +90 +105 +120 +i ( ) +0.50 +0.75 +1.00 +1.25 +Msec (M ) +25 +50 +75 +100 +125 +a (AU) +0.45 +0.60 +0.75 +e +75 +90 +105 +120 +i ( ) +i ( ) = 101.7+7.0 +8.8 +Fig. A.7: Corner plot of BD-11o3853 +Mpri (M ) = 2.69+0.60 +0.60 +0.6 +0.8 +1.0 +1.2 +Msec (M ) +Msec (M ) = 0.94+0.13 +0.14 +15 +18 +21 +24 +a (AU) +a (AU) = 21.1+1.6 +1.7 +0.05 +0.10 +0.15 +0.20 +e +e = 0.116+0.042 +0.041 +0.8 +1.6 +2.4 +3.2 +4.0 +Mpri (M ) +144 +146 +148 +150 +152 +i ( ) +0.6 +0.8 +1.0 +1.2 +Msec (M ) +15 +18 +21 +24 +a (AU) +0.05 +0.10 +0.15 +0.20 +e +144 +146 +148 +150 +152 +i ( ) +i ( ) = 147.8+1.5 +1.6 +Fig. A.8: Corner plot of HD 104979 +Article number, page 18 of 49 + +CORAVEL +HIPPARCOS +HIPPARCOS +-142 +-32 +7 1.2 +HERMES +GAIA +○ GAIA +2000 +-144 +-34 +1.1 +o上 +(uk/sew) +(μk/sew) +-146 +-2000 +-36 +1.0 +/w) +-148 +-4000 +*0nl +-38 +-150 +-6000 +-40 +-152 +-8000F +42 +0.7 +2500 +2.5 E +2.5 +0.6 +O-C +O-C +AA +0.0 +0.0 +-2500 +-2.5 +-2.5 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)2000 +60 E +HIPPARCOS +DAO +HIPPARCOS +-212.5 F +GAIA +CORAVEL +GAIA +1.2 +HERMES +55 +1000 +-215.0 +-217.5 +1.1 +(mas/yr) +50 +(k/se) +(s/w) +0 +-220.0E +45 +1.0 +-222.5 +-1000 +-225.0E +40 +0.9 +-227.5 F +-2000 +35上 +230.0 E +0.8 +882- +-232.5 E +0.7 +0.5 +0.5 +O-C +O-C +0.0 +0.0 +-0.5 +-0.5 +-2000 +1980 +2000 +1990 +2010 +2020 +1990 +1995 +2000 +2005 +20102015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (year) +Epoch (year) +Epoch (yr)A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. A.9: RV curve and proper motions of HD 51959. We used a fixed RV offset of 500 m/s (Jorissen et al. 2019). +Fig. A.10: RV curve and proper motions of HD 123949 +Mpri (M ) = 1.20+0.30 +0.30 +0.30 +0.45 +0.60 +0.75 +Msec (M ) +Msec (M ) = 0.509+0.077 +0.080 +7.5 +10.0 +12.5 +15.0 +17.5 +a (AU) +a (AU) = 11.75+0.88 +1.0 +0.1 +0.2 +0.3 +0.4 +e +e = 0.295+0.040 +0.047 +0.4 +0.8 +1.2 +1.6 +2.0 +Mpri (M ) +155.0 +157.5 +160.0 +162.5 +165.0 +i ( ) +0.30 +0.45 +0.60 +0.75 +Msec (M ) +7.5 +10.0 +12.5 +15.0 +17.5 +a (AU) +0.1 +0.2 +0.3 +0.4 +e +155.0 +157.5 +160.0 +162.5 +165.0 +i ( ) +i ( ) = 163.16+0.71 +0.82 +Fig. A.11: Corner plot of HD 51959 +Mpri (M ) = 1.55+0.50 +0.41 +0.6 +0.8 +1.0 +1.2 +Msec (M ) +Msec (M ) = 0.78+0.15 +0.13 +9 +10 +11 +12 +13 +a (AU) +a (AU) = 10.84+0.92 +0.91 +0.915 +0.916 +0.917 +0.918 +e +e = 0.91667+0.00065 +0.00066 +1.2 +1.8 +2.4 +3.0 +Mpri (M ) +60 +80 +100 +120 +i ( ) +0.6 +0.8 +1.0 +1.2 +Msec (M ) +9 +10 +11 +12 +13 +a (AU) +0.915 +0.916 +0.917 +0.918 +e +60 +80 +100 +120 +i ( ) +i ( ) = 122.4+4.5 +72 +Fig. A.12: Corner plot of HD 123949 +Article number, page 19 of 49 + +-24 +1500F +5 +0.65 +-26 +1000 +4 +μα* (mas/yr) +0.60 +(μk/sew) +RV (m/s) +3/ +500 +公 +-28 +0.55 +2 +1 +(Mo) +30 +-500 E +0.45 +-1000 +CORAVEL +HIPPARCOS +HIPPARCOS +32 +HERMES +GAIA + GAIA +0.40 +-2 +2 +1000 +0.35 +O-C +O-C +O-C +0 +0 +0 +-1000 +1990 +2000 +2010 +2020 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)18F +HIPPARCOS +7500 E +● GAIA +16F +1.1 +10 +5000 E +14E +2500 E +(uk/sew) +1.0 +000 +(mas/yr) +8 +(s/u) +12 +0 E +10 +0.9 +M +-2500 +6 +1comp(M。) +μα* +会 +8F +-5000 +0.8 +4 +6E +-7500 +4 +-10000E +CORAVEL +HIPPARCOS + 0.7 +2 +HERMES +GAIA +1000 +2.5 +0.6 +大 +O-C +O-C +0.0 +-1000 +-2.5 +1990 +2000 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +2010 +2020 +Epoch (yr) +Epoch (year) +Epoch (year)A&A proofs: manuscript no. main +Fig. A.13: RV curve and proper motions of HD 182274 +Fig. A.14: RV curve and proper motions of HD 18182. We used a fixed RV offset of 500 m/s (Jorissen et al. 2019). +Mpri (M ) = 1.10+0.20 +0.20 +0.3 +0.4 +0.5 +0.6 +0.7 +Msec (M ) +Msec (M ) = 0.549+0.060 +0.061 +7.2 +8.0 +8.8 +9.6 +10.4 +a (AU) +a (AU) = 9.55+0.47 +0.53 +0.025 +0.050 +0.075 +0.100 +e +e = 0.039+0.013 +0.011 +0.6 +0.9 +1.2 +1.5 +Mpri (M ) +48.0 +49.5 +51.0 +52.5 +i ( ) +0.3 +0.4 +0.5 +0.6 +0.7 +Msec (M ) +7.2 +8.0 +8.8 +9.6 +10.4 +a (AU) +0.025 +0.050 +0.075 +0.100 +e +48.0 +49.5 +51.0 +52.5 +i ( ) +i ( ) = 50.49+1.0 +0.93 +Fig. A.15: Corner plot of HD 182274 +Mpri (M ) = 1.79+0.62 +0.58 +4 +8 +12 +Msec (M ) +Msec (M ) = 0.35+0.36 +0.12 +30 +60 +90 +120 +150 +a (AU) +a (AU) = 10.9+8.7 +1.5 +0.2 +0.4 +0.6 +0.8 +e +e = 0.35+0.37 +0.22 +0.8 +1.6 +2.4 +3.2 +Mpri (M ) +40 +80 +120 +160 +i ( ) +4 +8 +12 +Msec (M ) +30 +60 +90 +120 +150 +a (AU) +0.2 +0.4 +0.6 +0.8 +e +40 +80 +120 +160 +i ( ) +i ( ) = 33+26 +13 +Fig. A.16: Corner plot of HD 18182 +Article number, page 20 of 49 + +4000 +10 +CORAVEL + SOPHIE +3000 +HERMES +5/ +-75 +0.65 +2000E +(uk/sew) +0 +1000E +(uk/sew) +(s/u) +-80 +0.60 +0 +-1000 +-85 +*0 +-10 +0.55 +-2000F +-90 +-3000 +0.50 +-20F +HIPPARCOS +HIPPARCOS +-4000 +GAIA + GAIA +.95 +0.45 +1000 +O-C +O-C +O-C +公 +0F +0.40 +-1000 +-1 E +1990 +2000 +2010 +2020 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)1500 +CORAVEL +HIPPARCOS +-8 +4.5 +HERMES +11.5 +GAIA +1000 +11.0 +4.0 +500 +-9 +μs (mas/yr) +3.5 +(s/w), +3.0 +-10 +9.5 +-500 +2.5 +9.0E +-11 +-1000 +2.0 +8.5 +1.5 +-1500 +8.0 +-12 +1.0 +-2000 +1000 +2 F +2.5F +0.5 +O-C +C +0 +0.0 +0 +0 +-1000 +1990 1995 2000 2005 2010 2015 +1990 +1995 +2000 2005 2010 2015 +1990 +2000 +2010 +2020 +Epoch (yr) +Epoch (year) +Epoch (year)A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. A.17: RV curve and proper motions of HD 53199. +Fig. A.18: RV curve and proper motions of HD 40430 +Mpri (M ) = 2.50+0.30 +0.30 +0.48 +0.56 +0.64 +0.72 +0.80 +Msec (M ) +Msec (M ) = 0.642+0.049 +0.051 +10.4 +11.2 +12.0 +12.8 +a (AU) +a (AU) = 11.66+0.44 +0.47 +0.225 +0.240 +0.255 +0.270 +e +e = 0.2549+0.0091 +0.010 +1.6 +2.0 +2.4 +2.8 +3.2 +Mpri (M ) +70 +80 +90 +100 +110 +i ( ) +0.48 +0.56 +0.64 +0.72 +0.80 +Msec (M ) +10.4 +11.2 +12.0 +12.8 +a (AU) +0.225 +0.240 +0.255 +0.270 +e +70 +80 +90 +100 +110 +i ( ) +i ( ) = 103.2+5.3 +7.0 +Fig. A.19: Corner plot of HD 53199 +Mpri (M ) = 2.30+0.60 +0.59 +0.4 +0.6 +0.8 +1.0 +Msec (M ) +Msec (M ) = 0.70+0.11 +0.12 +6.0 +7.5 +9.0 +10.5 +a (AU) +a (AU) = 9.44+0.74 +0.84 +0.21 +0.24 +0.27 +0.30 +e +e = 0.262+0.021 +0.025 +0.8 +1.6 +2.4 +3.2 +4.0 +Mpri (M ) +30 +60 +90 +120 +150 +i ( ) +0.4 +0.6 +0.8 +1.0 +Msec (M ) +6.0 +7.5 +9.0 +10.5 +a (AU) +0.21 +0.24 +0.27 +0.30 +e +30 +60 +90 +120 +150 +i ( ) +i ( ) = 23.36+0.58 +0.55 +Fig. A.20: Corner plot of HD 40430 +Article number, page 21 of 49 + +4000E +0.75 +-2 +-5.0F +3000 E +5.5 +2000 E +(mas/yr) +0.70 + (mas/yr) +-6.0 +RV (m/s) +1000 +-6.5 +-4 +oF +*on +0.65 +mp(Mo) +-7.0E +-1000F +5 +7.5 +-2000 +0.60 +CORAVEL +-8.0E +ELODIE +HIPPARCOS +HIPPARCOS +-3000 +HERMES +GAIA +GAIA +6 +2.5 F +0.55 +500 +O-C +0.0 +-500 +2.5 +1990 +2000 +2010 +2020 +1990 +1995 +2000 +2005 +2010 +2015 +19901995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)-2 +HIPPARCOS +-10F +2000 +GAIA +0.9 +1500E +-3 +-11 +1000F +μα* (mas/yr) +(mas/yr) +-12 +0.8 +RV (m/s) +-4F +500E +Mcomp(Mo) +-13 +0.7 +-5 +-500F +-14F +-1000 +0.6 +-15上 +-1500 +CORAVEL +HIPPARCOS +不 +HERMES +GAIA +0.5 +2.5 F +1000 +1F +0-0 +0-0 +O-C +0 +- +0.0 +0.4 +-1000 +-2.5 E. +1990 +2000 +2010 +2020 +2030 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 + 2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A&A proofs: manuscript no. main +Fig. A.21: RV curve and proper motions of HD 95241 +Fig. A.22: RV curve and proper motions of HD 139195 +Mpri (M ) = 1.52+1.0 +0.80 +0.15 +0.30 +0.45 +0.60 +Msec (M ) +Msec (M ) = 0.34+0.12 +0.12 +4.5 +6.0 +7.5 +9.0 +10.5 +a (AU) +a (AU) = 7.4+1.2 +1.5 +0.795 +0.800 +0.805 +0.810 +0.815 +e +e = 0.8071+0.0034 +0.0037 +1 +2 +3 +4 +Mpri (M ) +102 +105 +108 +111 +i ( ) +0.15 +0.30 +0.45 +0.60 +Msec (M ) +4.5 +6.0 +7.5 +9.0 +10.5 +a (AU) +0.795 +0.800 +0.805 +0.810 +0.815 +e +102 +105 +108 +111 +i ( ) +i ( ) = 107.7+1.8 +2.1 +Fig. A.23: Corner plot of HD 95241 +Mpri (M ) = 2.60+0.30 +0.30 +0.56 +0.64 +0.72 +0.80 +Msec (M ) +Msec (M ) = 0.662+0.048 +0.050 +8.0 +8.5 +9.0 +9.5 +a (AU) +a (AU) = 8.82+0.30 +0.33 +0.27 +0.30 +0.33 +0.36 +e +e = 0.318+0.022 +0.022 +2.0 +2.4 +2.8 +3.2 +Mpri (M ) +94.5 +96.0 +97.5 +99.0 +100.5 +i ( ) +0.56 +0.64 +0.72 +0.80 +Msec (M ) +8.0 +8.5 +9.0 +9.5 +a (AU) +0.27 +0.30 +0.33 +0.36 +e +94.5 +96.0 +97.5 +99.0 +100.5 +i ( ) +i ( ) = 97.6+1.1 +1.1 +Fig. A.24: Corner plot of HD 139195 +Article number, page 22 of 49 + +HIPPARCOS +75 E +0.6 + GAIA +-120 +2000 +-80 F +125E + 0.5 +(u/sew) +-85 +(mas/yr) +oF +(s/w) +-130 +-90 +Mcomp(Mo) +0.42 +-135 +*0n +-95 +AAKA +-140 +-100E +-4000 +0.3 +CORAVEL +OIK +-105E +-145E +HERMES +HIPPARCOS +SOPHIE +GAIA +0.2 +1000 +1E +O-C +O-C +O-C +0 +W- - +0 +-1000 +-1E +2000 +2010 +2020 +2031 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)8000 +Lick +HIPPARCOS +HIPPARCOS +48 上 +GAIA +GAIA + Camb, +115 + DAO +0.75 +6000F +CORAVEL +46 +HERMES +44 +(mas/yr) +-120 +4000上 +μs (mas/yr) +RV (m/s) +42 E +0.70 +2000上 +Mcomp(Mo) +-125 +40F +*n +oF +38上 +0.65 +-130 +-2000 +36 F +34 +-135 +-4000 +0.60 +5000 +O-C +O-C +0-0 +0 +0 +0.55 +-5000. +2 +1920 +1940 +1960 +1980 +2000 +2020 +1990 +1995 +2000 +2005 +20102015 +1990 +1995 +2000 +2005 + 2010 +2015 +Epoch (year) +Epoch (yr) +Epoch (year)A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. A.25: RV curve and proper motions of BD-10o4311 +Fig. A.26: RV curve and proper motions of HD 183915 +Mpri (M ) = 0.79+0.29 +0.29 +0.15 +0.30 +0.45 +0.60 +0.75 +Msec (M ) +Msec (M ) = 0.52+0.10 +0.11 +4 +5 +6 +7 +a (AU) +a (AU) = 6.15+0.56 +0.72 +0.03 +0.04 +0.05 +0.06 +0.07 +e +e = 0.0470+0.0062 +0.0058 +0.4 +0.8 +1.2 +1.6 +Mpri (M ) +70 +80 +90 +100 +110 +i ( ) +0.15 +0.30 +0.45 +0.60 +0.75 +Msec (M ) +4 +5 +6 +7 +a (AU) +0.03 +0.04 +0.05 +0.06 +0.07 +e +70 +80 +90 +100 +110 +i ( ) +i ( ) = 72.5+3.2 +2.5 +Fig. A.27: Corner plot of BD-10o4311 +Mpri (M ) = 1.87+1.0 +0.88 +0.2 +0.4 +0.6 +0.8 +1.0 +Msec (M ) +Msec (M ) = 0.61+0.17 +0.19 +4.5 +6.0 +7.5 +9.0 +a (AU) +a (AU) = 7.05+0.94 +1.2 +0.30 +0.35 +0.40 +0.45 +0.50 +e +e = 0.408+0.036 +0.035 +1 +2 +3 +4 +5 +Mpri (M ) +173.6 +174.0 +174.4 +174.8 +i ( ) +0.2 +0.4 +0.6 +0.8 +1.0 +Msec (M ) +4.5 +6.0 +7.5 +9.0 +a (AU) +0.30 +0.35 +0.40 +0.45 +0.50 +e +173.6 +174.0 +174.4 +174.8 +i ( ) +i ( ) = 174.25+0.28 +0.29 +Fig. A.28: Corner plot of HD 183915 +Article number, page 23 of 49 + +CORAVEL +-72 +6000 +CORALIE +HERMES +-46 + 0.7 +-74 +4000 +-76 +-48 +2000 +(mas/yr) +(ak/sew) +RV (m/s) +-78 +0.6 +-50 +oF +-80 +-52 +-2000F +0.5 +-82 +-4000 +-84 +-54 +0.4 +-6000F +-86 +HIPPARCOS +-56 +HIPPARCOS +GAIA +GAIA +-88 +2500 +2.5 +0.3 +2.5 +O-C +0-0 +A +0.0 +0.0 +0 +-2.5 +2.5 +-2500 +1995 +1990 +2000 +2010 +2020 +2030 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)1500 +DAO +8E + CORAVEL +HERMES +1000F +0.9 +7 +μα* (mas/yr) +500E + (mas/yr) +0.8 +6 +RV (m/s) +0.7 +0 +2 +0.6 +4 E +-500 +1 +0.5 +3 E +-1000F +HIPPARCOS +0.4 +GAIA +2 +2000 +2 F + 0.3 +O-C +0-0 +O-C +0 +0 +0 +-2000 E +1980 +1990 +2000 +2010 +2020 +2030 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A&A proofs: manuscript no. main +Fig. A.29: RV curve and proper motions of HD 180622. We used a fixed RV offset of 500 m/s (Jorissen et al. 2019). +Fig. A.30: RV curve and proper motions of HD 216219. We used a fixed HERMES-CORAVEL RV offset of 186 m/s Escorza et al. +(2019b) +Mpri (M ) = 1.81+0.88 +0.85 +0.3 +0.6 +0.9 +1.2 +Msec (M ) +Msec (M ) = 0.80+0.21 +0.24 +3.0 +4.5 +6.0 +7.5 +9.0 +a (AU) +a (AU) = 6.85+0.85 +1.1 +0.06 +0.12 +0.18 +0.24 +0.30 +e +e = 0.079+0.033 +0.035 +1 +2 +3 +4 +Mpri (M ) +88 +96 +104 +112 +i ( ) +0.3 +0.6 +0.9 +1.2 +Msec (M ) +3.0 +4.5 +6.0 +7.5 +9.0 +a (AU) +0.06 +0.12 +0.18 +0.24 +0.30 +e +88 +96 +104 +112 +i ( ) +i ( ) = 100.1+5.5 +5.3 +Fig. A.31: Corner plot of HD 180622 +Mpri (M ) = 1.45+0.30 +0.30 +0.45 +0.60 +0.75 +Msec (M ) +Msec (M ) = 0.631+0.077 +0.081 +6 +8 +10 +12 +14 +a (AU) +a (AU) = 6.24+0.35 +0.40 +0.15 +0.30 +0.45 +e +e = 0.085+0.048 +0.047 +0.8 +1.2 +1.6 +2.0 +2.4 +Mpri (M ) +50 +75 +100 +125 +i ( ) +0.45 +0.60 +0.75 +Msec (M ) +6 +8 +10 +12 +14 +a (AU) +0.15 +0.30 +0.45 +e +50 +75 +100 +125 +i ( ) +i ( ) = 33.5+1.8 +1.6 +Fig. A.32: Corner plot of HD 216219 +Article number, page 24 of 49 + +6000F +CORAVEL +HIPPARCOS +HERMES +16 +GAIA +-12E +1.2 +4000 +-18 +-13 +2000 +(mas/yr) +(mas/yr) +-20 +1.0 +RV (m/s) +-14 +oF +-22 +*n +-15 +0.8 +5 +2000 +-24上 +16 +-4000 +26 +0.6 +HIPPARCOS +-17 +GAIA +28 +-6000 +0.4 +1000 +O-C +0-0 +-1000 +1990 +2000 +2010 +2020 +2030 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)HIPPARCOS +57.5 +GAIA +GAIA +3000 +0.80 +20 A +55.0E +2000 +0.75 +(μK/sew) *r +52.5 +1000 +μs (mas/yr) +RV (m/s) +15 +0.70 +0 +47.5 E +0.65 +10F +-1000 +45.0 E +0.60 +-2000 +5 +42.5 +-3000 +0.55 +CORAVEL +40.0 +HERMES +0.50 +2000 +2.5 +2.5 F +O-C +0-0 +O-C +0.0 +0.0 +0.45 +-2000 +2.5 +-2.5 +1980 +1990 +2000 +2010 +2020 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. A.33: RV curve and proper motions of HD 107541 +Fig. A.34: RV curve and proper motions of BD-14o2678 +Mpri (M ) = 1.13+0.57 +0.50 +0.2 +0.4 +0.6 +0.8 +1.0 +Msec (M ) +Msec (M ) = 0.55+0.15 +0.16 +4 +5 +6 +7 +a (AU) +a (AU) = 5.46+0.69 +0.83 +0.05 +0.10 +0.15 +0.20 +e +e = 0.095+0.034 +0.036 +0.6 +1.2 +1.8 +2.4 +3.0 +Mpri (M ) +50 +75 +100 +125 +i ( ) +0.2 +0.4 +0.6 +0.8 +1.0 +Msec (M ) +4 +5 +6 +7 +a (AU) +0.05 +0.10 +0.15 +0.20 +e +50 +75 +100 +125 +i ( ) +i ( ) = 127.5+3.0 +3.1 +Fig. A.35: Corner plot of HD 107541 +Mpri (M ) = 3.00+0.60 +0.60 +0.45 +0.60 +0.75 +0.90 +1.05 +Msec (M ) +Msec (M ) = 0.671+0.10 +0.094 +5.6 +6.4 +7.2 +8.0 +a (AU) +a (AU) = 6.97+0.46 +0.50 +0.16 +0.24 +0.32 +0.40 +e +e = 0.253+0.045 +0.049 +1.6 +2.4 +3.2 +4.0 +4.8 +Mpri (M ) +60 +80 +100 +120 +i ( ) +0.45 +0.60 +0.75 +0.90 +1.05 +Msec (M ) +5.6 +6.4 +7.2 +8.0 +a (AU) +0.16 +0.24 +0.32 +0.40 +e +60 +80 +100 +120 +i ( ) +i ( ) = 93+16 +18 +Fig. A.36: Corner plot of BD-14o2678 +Article number, page 25 of 49 + +-26 +CORAVEL +HIPPARCOS +4000 +GAIA +0.9 +-14 +-28 +0.8 +2000 +-16 +(μk/sew) +(k/sew) 9r +RV (m/s) +0.7 +-30 +-18 +*on +0.6 +-2000 +-32 +20 +0.5 +-22 +-4000 +-34 +0.4 +1000 +2.5 +O-C +0.3 +O-C +O-C +0.0 F +-1000 +-2.5 +198619881990 +1990 +1995 +2000 +1992 +1994 +1996 +2005 + 2010 + 2015 +1990 +1995 +2000 +2005 +20102015 +Epoch (yr) +Epoch (year) +Epoch (year)Φ CORAVEL +HIPPARCOS +-4.5上 +GAIA +0.9 +4000 +1.5 +5.0 +W +1.0 +2000 +(u/sew) +-5.5 +0.8 +RV (m/s) +0.5 E +-6.0 E +0 +*on +0.7 +-6.5 +0.0 +-2000 +7.0 +-0.5 E +0.6 +HIPPARCOS +7.5 +-4000 +GAIA +-1.0E +1000 +0.5 +O-C +C +-1E +-1000 +1990　1995 +20002005 + 20102015 +1990 +1995 +20002005 +2010 2015 +1988 +1990 +1992 +1994 +1996 +Epoch (yr) +Epoch (year) +Epoch (year)A&A proofs: manuscript no. main +Fig. A.37: RV curve and proper motions of HD 59852 +Fig. A.38: RV curve and proper motions of HD 201824. We used a fixed RV offset of 500 m/s (Jorissen et al. 2019). +Mpri (M ) = 2.51+0.89 +0.87 +0.4 +0.6 +0.8 +1.0 +Msec (M ) +Msec (M ) = 0.62+0.14 +0.15 +4 +5 +6 +7 +8 +a (AU) +a (AU) = 6.57+0.66 +0.81 +0.08 +0.16 +0.24 +0.32 +e +e = 0.141+0.078 +0.083 +1.5 +3.0 +4.5 +Mpri (M ) +20 +25 +30 +35 +i ( ) +0.4 +0.6 +0.8 +1.0 +Msec (M ) +4 +5 +6 +7 +8 +a (AU) +0.08 +0.16 +0.24 +0.32 +e +20 +25 +30 +35 +i ( ) +i ( ) = 25.8+3.6 +3.2 +Fig. A.39: Corner plot of HD 59852 +Mpri (M ) = 1.8+1.1 +1.0 +0.3 +0.6 +0.9 +1.2 +1.5 +Msec (M ) +Msec (M ) = 0.78+0.27 +0.28 +3 +4 +5 +6 +7 +a (AU) +a (AU) = 5.49+0.85 +1.1 +0.20 +0.25 +0.30 +0.35 +0.40 +e +e = 0.296+0.035 +0.034 +1.5 +3.0 +4.5 +Mpri (M ) +60 +80 +100 +120 +i ( ) +0.3 +0.6 +0.9 +1.2 +1.5 +Msec (M ) +3 +4 +5 +6 +7 +a (AU) +0.20 +0.25 +0.30 +0.35 +0.40 +e +60 +80 +100 +120 +i ( ) +i ( ) = 59.1+66 +6.3 +Fig. A.40: Corner plot of HD 201824 +Article number, page 26 of 49 + +CORAVEL +HIPPARCOS +-4.0 +2000 +GAIA +-4.5 +0.9 +3 +-5.0 +1000 +μα* (mas/yr) +0.8 +(mas/yr) +-5.5 +RV (m/s) +2 +-6.0 +0 +0.7 +-6.5 +0.6 +-1000 +-7.0 +-7.5 +70 +0.5 +-2000 +-8.0 +0.4 +1000 +2.5 F +O-C +O-C +0.0 +0.0 +0.3 +H +-1000 +2.5 +2.5 +1990 +1990 +1995 +2000 +2005 +2010 +2015 +1988 +1992 +1994 +1996 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)15 +24F +HIPPARCOS +HIPPARCOS +GAIA +GAIA +6000F +Q +14E +22 +4000 +1.2 +13 +8 +* (mas/yr) +(u/sew) +RV (m/s) +2000 +20 +12 +1.0 +11 +*0n +18 +-2000 +0.8 +10 +-4000 +16 +9 +0.6 +-6000 +CORAVEL +HERMES +10000 F +0.4 +1 +O-C +O-C +O-C +0F +C +0 +-10000 +.5 +2015 +1995 +1980 +1990 +2000 +2010 +2020 +1990 +2000 +2005 +2010 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. A.41: RV curve and proper motions of HD 178717. We used a fixed RV offset of 500 m/s (Jorissen et al. 2019). +Fig. A.42: RV curve and proper motions of HD 50082. We used a fixed RV offset of 500 m/s (Jorissen et al. 2019). +Mpri (M ) = 1.73+0.81 +0.77 +0.2 +0.4 +0.6 +0.8 +1.0 +Msec (M ) +Msec (M ) = 0.53+0.14 +0.16 +3 +4 +5 +6 +7 +a (AU) +a (AU) = 5.24+0.65 +0.84 +0.40 +0.44 +0.48 +0.52 +e +e = 0.461+0.027 +0.028 +1 +2 +3 +4 +Mpri (M ) +30 +60 +90 +120 +150 +i ( ) +0.2 +0.4 +0.6 +0.8 +1.0 +Msec (M ) +3 +4 +5 +6 +7 +a (AU) +0.40 +0.44 +0.48 +0.52 +e +30 +60 +90 +120 +150 +i ( ) +i ( ) = 34.7+4.2 +2.9 +Fig. A.43: Corner plot of HD 178717 +Mpri (M ) = 1.60+0.89 +0.78 +0.2 +0.4 +0.6 +0.8 +1.0 +Msec (M ) +Msec (M ) = 0.56+0.17 +0.18 +3 +4 +5 +6 +7 +a (AU) +a (AU) = 5.13+0.73 +0.91 +0.15 +0.18 +0.21 +0.24 +e +e = 0.189+0.020 +0.020 +1 +2 +3 +4 +Mpri (M ) +56 +60 +64 +68 +72 +i ( ) +0.2 +0.4 +0.6 +0.8 +1.0 +Msec (M ) +3 +4 +5 +6 +7 +a (AU) +0.15 +0.18 +0.21 +0.24 +e +56 +60 +64 +68 +72 +i ( ) +i ( ) = 63.2+2.5 +2.2 +Fig. A.44: Corner plot of HD 50082 +Article number, page 27 of 49 + +4000 +CORAVEL +12 E +OIK +HIPPARCOS +HERMES +GAIA +10 F +3000 +11 +0.8 +9 +2000 + (mas/yr) +10E +0.7 +(mas/yr) +8 +1000 +RV (m/s) +E +16 +7 +0.6 +oE +Mcomp(Mo) +*on +μs +6 +8E +-1000 +0.5 +5 +7 +0.4 +-3000 +6 +0.3 +2500F +0.5 +O-C +0.2 +O-C +O-C +0.0 +一 +-0.5 +2500Ei +2005 +2015 +1990 +1980 +1990 +2000 +2010 +2020 +1990 +1995 +2000 +2010 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)14 F +4000 +13 +0.9 +12 +2000 +0.8 +μα* (mas/yr) +6 +μs (mas/yr) +11 +RV (m/s) +0.7 +0 +HIPPARCOS +10F +GAIA +4 +0.6 +-2000 +9E +0.5 +8 E +-4000 +7 E +0.4 +D +CORAVEL +HIPPARCOS +-6000 +HERMES +GAIA +0.3 +1000 +O-C +O-0 +0 +O +0.2 +-1000 +1980 +1990 +2000 +2010 +2020 +1990 +1995 +2000 +2005 + 2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A&A proofs: manuscript no. main +Fig. A.45: RV curve and proper motions of HD 98991 +Fig. A.46: RV curve and proper motions of HD 205011 +Mpri (M ) = 1.43+0.30 +0.31 +0.30 +0.45 +0.60 +0.75 +Msec (M ) +Msec (M ) = 0.568+0.068 +0.076 +3.6 +4.2 +4.8 +5.4 +a (AU) +a (AU) = 4.96+0.29 +0.34 +0.312 +0.318 +0.324 +0.330 +0.336 +e +e = 0.3234+0.0038 +0.0039 +0.5 +1.0 +1.5 +2.0 +Mpri (M ) +123 +124 +125 +126 +127 +i ( ) +0.30 +0.45 +0.60 +0.75 +Msec (M ) +3.6 +4.2 +4.8 +5.4 +a (AU) +0.312 +0.318 +0.324 +0.330 +0.336 +e +123 +124 +125 +126 +127 +i ( ) +i ( ) = 124.88+0.72 +0.74 +Fig. A.47: Corner plot of HD 98991 +Mpri (M ) = 1.79+0.90 +0.83 +0.4 +0.6 +0.8 +1.0 +Msec (M ) +Msec (M ) = 0.61+0.17 +0.19 +3.2 +4.0 +4.8 +5.6 +6.4 +a (AU) +a (AU) = 5.26+0.69 +0.88 +0.16 +0.20 +0.24 +0.28 +e +e = 0.231+0.024 +0.024 +1 +2 +3 +4 +Mpri (M ) +66 +72 +78 +84 +90 +i ( ) +0.4 +0.6 +0.8 +1.0 +Msec (M ) +3.2 +4.0 +4.8 +5.6 +6.4 +a (AU) +0.16 +0.20 +0.24 +0.28 +e +66 +72 +78 +84 +90 +i ( ) +i ( ) = 74.1+4.6 +4.2 +Fig. A.48: Corner plot of HD 205011 +Article number, page 28 of 49 + +-290 +CORAVEL +6000 +HERMES +-20 +0.70 +-300 +4000 +0.65 +(mas/yr) +-30 + (mas/yr) +RV (m/s) +KAAK +-310 +2000 +0.60 +HIPPARCOS + GAIA +-40 +-320 +μα* +0 +0.55 +330 +50 +-2000 +0.50 +0.45 +-4000 +340 +-60 +2500 +2.5 +0.40 +O-C +0-0 +O-C +0 +0.0 +A +一 +-2500 +-2.5 E +1990 +1995 +20002005　2010　2015 +1995 +2000 +2005 +2010 +2015 +2020 +1990 +1995 +20002005 + 20102015 +Epoch (yr) +Epoch (year) +Epoch (year)6000 +每口 +10 +0.9 +4000 +8 +(μ/sew) * +4 +0.8 +2000 +μs (mas/yr) +RV (m/s) +6 +0.7 +0 +2 +Mcomp(Mo) +4 +*on +0.6 +-2000 +2 +0 +0.5 +-4000 +0 +HIPPARCOS +HIPPARCOS +DAO +一 +HERMES +GAIA +GAIA +0.4 +2500 +O-C +O-C +O-C +0 +0.3 +-2500 +1975 +1980 1985 +1990 1995 +2000 2005 +2010 +1995 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. A.49: RV curve and proper motions of HD 204075 +Fig. A.50: RV curve and proper motions of HD 20394. We used a fixed RV offset of 500 m/s (Jorissen et al. 2019). +Mpri (M ) = 4.49+0.90 +0.90 +0.45 +0.60 +0.75 +0.90 +Msec (M ) +Msec (M ) = 0.67+0.10 +0.10 +4.8 +5.4 +6.0 +6.6 +a (AU) +a (AU) = 6.01+0.36 +0.41 +0.16 +0.24 +0.32 +0.40 +e +e = 0.257+0.049 +0.049 +3.0 +4.5 +6.0 +Mpri (M ) +120 +128 +136 +144 +i ( ) +0.45 +0.60 +0.75 +0.90 +Msec (M ) +4.8 +5.4 +6.0 +6.6 +a (AU) +0.16 +0.24 +0.32 +0.40 +e +120 +128 +136 +144 +i ( ) +i ( ) = 133.2+4.2 +4.5 +Fig. A.51: Corner plot of HD 204075 +Mpri (M ) = 2.01+0.61 +0.59 +0.30 +0.45 +0.60 +0.75 +Msec (M ) +Msec (M ) = 0.491+0.095 +0.10 +3.0 +3.6 +4.2 +4.8 +5.4 +a (AU) +a (AU) = 4.56+0.39 +0.46 +0.08 +0.16 +0.24 +0.32 +e +e = 0.161+0.057 +0.057 +0.8 +1.6 +2.4 +3.2 +Mpri (M ) +30 +60 +90 +120 +150 +i ( ) +0.30 +0.45 +0.60 +0.75 +Msec (M ) +3.0 +3.6 +4.2 +4.8 +5.4 +a (AU) +0.08 +0.16 +0.24 +0.32 +e +30 +60 +90 +120 +150 +i ( ) +i ( ) = 30.8+2.5 +2.1 +Fig. A.52: Corner plot of HD 20394 +Article number, page 29 of 49 + +DAO +HIPPARCOS +HIPPARCOS +3000F +HERMES +4 F +GAIA +28 +GAIA +2000E +26F +2 +0.8 +1000 +(mas/yr) +μs (mas/yr) +24 E +RV (m/s) +0 +22 [ +0.7 +-2 +-1000 +on +20F +-2000 +4 +18 +0.6 +3000 +16 +-6 +2500 +2.5 +2.5 F +0.5 +O-C +O-C +O-C +0.0 +0 +口 +0.0 +2.5 E +2500 +2.5 +19952000200520102015 +19901995200020052010 2015 +1980 +1985 +1990 +1995 +2000 +2005 +2010 +1990 +Epoch (yr) +Epoch (year) +Epoch (year)CORAVEL +3000 +HIPPARCOS +HIPPARCOS +HERMES +GAIA +GAIA +3/ +0.7 +4 +2000 +-5 +2 E +1000 +μα* (mas/yr) +μs (mas/yr) +0.6 +1 +(s/w) +o[ +RV +-1000 +7 +0.5 +-1 +-2000 +-8E +-3000 +-2 +-9F +0.4 +-4000 +-3 +2500 +2.5 F +1F +0.3 +O-C +O-C +O-C +0.0 +0 +-2500 +2.5 E +1990 +1995 +2000 +2005 + 2010 +2015 +19901995 +20002005 +1985 +1990 +1995 +2000 +2005 + 2010 2015 2020 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A&A proofs: manuscript no. main +Fig. A.53: RV curve and proper motions of HD 16458 +Fig. A.54: RV curve and proper motions of HD 5424. We used a fixed RV offset of 500 m/s (Jorissen et al. 2019). +Mpri (M ) = 1.75+0.58 +0.56 +0.4 +0.6 +0.8 +1.0 +1.2 +Msec (M ) +Msec (M ) = 0.74+0.15 +0.15 +3.0 +3.5 +4.0 +4.5 +5.0 +a (AU) +a (AU) = 4.24+0.37 +0.45 +0.04 +0.08 +0.12 +0.16 +e +e = 0.098+0.025 +0.025 +0.8 +1.6 +2.4 +3.2 +Mpri (M ) +60 +80 +100 +120 +i ( ) +0.4 +0.6 +0.8 +1.0 +1.2 +Msec (M ) +3.0 +3.5 +4.0 +4.5 +5.0 +a (AU) +0.04 +0.08 +0.12 +0.16 +e +60 +80 +100 +120 +i ( ) +i ( ) = 60.7+12 +5.8 +Fig. A.55: Corner plot of HD 16458 +Mpri (M ) = 1.30+0.40 +0.39 +0.30 +0.45 +0.60 +0.75 +0.90 +Msec (M ) +Msec (M ) = 0.52+0.11 +0.11 +2.5 +3.0 +3.5 +4.0 +4.5 +a (AU) +a (AU) = 3.69+0.31 +0.37 +0.08 +0.16 +0.24 +0.32 +e +e = 0.188+0.054 +0.052 +0.5 +1.0 +1.5 +2.0 +2.5 +Mpri (M ) +30 +60 +90 +120 +150 +i ( ) +0.30 +0.45 +0.60 +0.75 +0.90 +Msec (M ) +2.5 +3.0 +3.5 +4.0 +4.5 +a (AU) +0.08 +0.16 +0.24 +0.32 +e +30 +60 +90 +120 +150 +i ( ) +i ( ) = 30.2+3.1 +2.8 +Fig. A.56: Corner plot of HD 5424 +Article number, page 30 of 49 + +6000 +HIPPARCOS +HIPPARCOS +1.1 +GAIA +-65 +GAIA +4000F +20 +1.0 +2000 +70 +μα* (mas/yr) +μs (mas/yr) +RV (m/s) +0.9 +15 +0.8 +-2000 +4000 +-80 +0.7 +-6000 +5 +0.6 +CORAVEL +-85 +-8000 +2.5 +2.5 +0.5 +1000 +O-C +0-0 +O-C +0 +0.0 +0.0 +-1000 +2.5 +-2.5 +1980 +1982 +1984 +1986 +1988 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)-8 +CORAVEL +32 +HIPPARCOS +HIPPARCOS +3000 +GAIA +HERMES + GAIA +-33 +2000 +0.7 +-34 +1000 +(mas/yr) +-11 +(uK/sew) 9r +RV (m/s) +-12 +-35 +0.6 +Mcomp(Mo) +*on +13 +-1000 +-36 +-14 +0.5 +-2000 +-37 +-15 +-3000 +0.4 +16 +2.5 +500 +O-C +O-C +0.3 +0.0 +-500 +2.5 +1995 +1990 1995 2000 2005 + 2010 2015 +2020 +1990 +2000 +2005 + 2010 + 2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. A.57: RV curve and proper motions of HD 49641 +Fig. A.58: RV curve and proper motions of HD 91208. We used a fixed RV offset of 500 m/s (Jorissen et al. 2019). +Mpri (M ) = 3.7+2.1 +1.8 +1 +2 +3 +4 +Msec (M ) +Msec (M ) = 1.23+0.39 +0.41 +4 +8 +12 +16 +a (AU) +a (AU) = 4.91+0.75 +0.89 +0.2 +0.4 +0.6 +0.8 +e +e = 0.058+0.061 +0.041 +4 +8 +12 +16 +Mpri (M ) +40 +80 +120 +160 +i ( ) +1 +2 +3 +4 +Msec (M ) +4 +8 +12 +16 +a (AU) +0.2 +0.4 +0.6 +0.8 +e +40 +80 +120 +160 +i ( ) +i ( ) = 159.5+1.6 +1.6 +Fig. A.59: Corner plot of HD 49641 +Mpri (M ) = 2.29+0.60 +0.60 +0.4 +0.6 +0.8 +1.0 +1.2 +Msec (M ) +Msec (M ) = 0.83+0.13 +0.14 +3.0 +3.5 +4.0 +4.5 +5.0 +a (AU) +a (AU) = 4.19+0.30 +0.36 +0.12 +0.15 +0.18 +0.21 +0.24 +e +e = 0.178+0.018 +0.017 +0.8 +1.6 +2.4 +3.2 +4.0 +Mpri (M ) +124 +128 +132 +136 +140 +i ( ) +0.4 +0.6 +0.8 +1.0 +1.2 +Msec (M ) +3.0 +3.5 +4.0 +4.5 +5.0 +a (AU) +0.12 +0.15 +0.18 +0.21 +0.24 +e +124 +128 +132 +136 +140 +i ( ) +i ( ) = 133.6+2.1 +2.3 +Fig. A.60: Corner plot of HD 91208 +Article number, page 31 of 49 + +-3 +HIPPARCOS +HIPPARCOS +3000上 +-8 E +GAIA + GAIA +-9 +2.25 +2000 +-10 +2.00 +(mas/yr) +1000F +μs (mas/yr) +-6 +RV (m/s) +1.75 +12 +*on +-8 +mp(M。) +1.50 +-1000F +13 +-9 +-14 +1.25 +-2000F +10 +-15 +-3000 +1.00 +2500 +0.75 +0-0 +百 +O-C +O-C +古 +0.50 +-2500 +1995 +2005 +2015 +2005 +1980 +1982 +1984 +1986 +1988 +1990 +2000 +2010 +1990 +1995 +2000 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)6000 +CORAVEL +-30 +HIPPARCOS +HIPPARCOS +10 +HO +HERMES +GAIA +GAIA +1.1 +-32 +8 +1.0 +(mas/yr) +2000 +μs (mas/yr) +RV (m/s) +-34 +4 E +0.9 +Mcomp(Mo) +0 +*on +-36 +2 E +0.8 +-2000 +oF +-38 +-4000 +0.7 +-2上 +0.6 +2 +1000 +2.5 +O-C +O-C +O-C +0 +0.0 +-2.5 +0.5 +-1000 +-2 +199019952000200520102015 +1990 +1995 +2000 +¥2005 +1990 +1995 +2000 2005 +2010 +20152020 +20102015 +Epoch (yr) +Epoch (year) +Epoch (year)A&A proofs: manuscript no. main +Fig. A.61: RV curve and proper motions of HD 200063. We used a fixed RV offset of 500 m/s (Jorissen et al. 2019). +Fig. A.62: RV curve and proper motions of HD 43389. We used a fixed RV offset of 500 m/s (Jorissen et al. 2019). +Mpri (M ) = 2.44+1.2 +0.90 +0.6 +0.9 +1.2 +1.5 +Msec (M ) +Msec (M ) = 0.95+0.26 +0.22 +3.2 +4.0 +4.8 +5.6 +a (AU) +a (AU) = 4.26+0.52 +0.53 +0.05 +0.10 +0.15 +0.20 +e +e = 0.073+0.039 +0.039 +1.5 +3.0 +4.5 +6.0 +Mpri (M ) +60 +75 +90 +105 +120 +i ( ) +0.6 +0.9 +1.2 +1.5 +Msec (M ) +3.2 +4.0 +4.8 +5.6 +a (AU) +0.05 +0.10 +0.15 +0.20 +e +60 +75 +90 +105 +120 +i ( ) +i ( ) = 114.7+4.4 +47 +Fig. A.63: Corner plot of HD 200063 +Mpri (M ) = 2.2+1.2 +1.1 +0.3 +0.6 +0.9 +1.2 +1.5 +Msec (M ) +Msec (M ) = 0.76+0.25 +0.25 +2.4 +3.2 +4.0 +4.8 +5.6 +a (AU) +a (AU) = 3.96+0.58 +0.70 +0.05 +0.10 +0.15 +0.20 +e +e = 0.083+0.017 +0.017 +1.5 +3.0 +4.5 +6.0 +Mpri (M ) +60 +75 +90 +105 +i ( ) +0.3 +0.6 +0.9 +1.2 +1.5 +Msec (M ) +2.4 +3.2 +4.0 +4.8 +5.6 +a (AU) +0.05 +0.10 +0.15 +0.20 +e +60 +75 +90 +105 +i ( ) +i ( ) = 110.9+3.0 +30 +Fig. A.64: Corner plot of HD 43389 +Article number, page 32 of 49 + +5 +CORAVEL +6000 +HERMES +16 +4 +15 +1.4 +4000 +3 +2000 +μα* (mas/yr) +14 +(mas/yr) +RV (m/s) +1.2 +13 +0 +2 +-2000 +2 +1.0 +4000 +11 +0 +-6000 +10 F +0.8 +HIPPARCOS +HIPPARCOS +GAIA +GAIA +-8000 +9 +1E +2.5 +500 +0.6 +O-C +O-C +O-C +0 +0.0 +0 +上 +V +-500 +-2.5E +工 +1F +1980 +1990 +2000 +2010 +2020 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)HIPPARCOS +HIPPARCOS +6000 +GAIA +GAIA +-15 +-8 +4000 +-16 +1.2 +-9 +2000 +(mas/yr) +(mas/yr) +RV (m/s) +-10 +-17 +1.0 +0 +-18 +Mcomp(Mo) +-11 +-2000[ + *n +μ +0.8 +-12 +4000F +-20 E +-13 +-6000 +0.6 +CORAVEL +HERMES +-14 +1000 +2.5 +0.4 +O-C +O-C +0.0 +-1000 +-1日 +2.5 +2000 +2020 +2015 +2010 +1990 +2010 +1990 +1995 +2000 +2005 +2010 +1990 +1995 +2000 +2005 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. A.65: RV curve and proper motions of HD 27271. +Fig. A.66: RV curve and proper motions of HD 95193. We used a fixed RV offset of 500 m/s (Jorissen et al. 2019). +Mpri (M ) = 2.88+0.58 +0.56 +0.45 +0.60 +0.75 +0.90 +Msec (M ) +Msec (M ) = 0.700+0.086 +0.089 +3.6 +4.0 +4.4 +4.8 +a (AU) +a (AU) = 4.23+0.25 +0.27 +0.18 +0.20 +0.22 +0.24 +e +e = 0.2236+0.0069 +0.0072 +1.6 +2.4 +3.2 +4.0 +Mpri (M ) +60 +75 +90 +105 +i ( ) +0.45 +0.60 +0.75 +0.90 +Msec (M ) +3.6 +4.0 +4.4 +4.8 +a (AU) +0.18 +0.20 +0.22 +0.24 +e +60 +75 +90 +105 +i ( ) +i ( ) = 102.7+4.3 +5.1 +Fig. A.67: Corner plot of HD 27271 +Mpri (M ) = 2.70+0.30 +0.30 +0.6 +0.7 +0.8 +0.9 +Msec (M ) +Msec (M ) = 0.707+0.075 +0.062 +3.6 +3.8 +4.0 +4.2 +4.4 +a (AU) +a (AU) = 4.12+0.14 +0.15 +0.09 +0.12 +0.15 +0.18 +e +e = 0.133+0.021 +0.020 +2.0 +2.4 +2.8 +3.2 +3.6 +Mpri (M ) +60 +80 +100 +120 +i ( ) +0.6 +0.7 +0.8 +0.9 +Msec (M ) +3.6 +3.8 +4.0 +4.2 +4.4 +a (AU) +0.09 +0.12 +0.15 +0.18 +e +60 +80 +100 +120 +i ( ) +i ( ) = 81+25 +17 +Fig. A.68: Corner plot of HD 95193 +Article number, page 33 of 49 + +HIPPARCOS +HIPPARCOS +GAIA +GAIA +-56 +4000 +-1 F +0.85 +-58 +2000 +-2 +μα* (mas/yr) +0.80 +μs (mas/yr) +RV (m/s) +-60 +0 +-3 +0.75 +-62 +-2000 +0.70 +-4 +-64 +-4000 +0.65 +5 +-66 +-6000 +CORAVEL +VO +HERMES +0.60 +SOPHIE +-68 +-6 +1000 +0.55 +2.5 +O-C +O-C +O-C +0 +0.0 +0.0 +0.50 +2.5 E +-1000 +2000 +2020 +1990 +1995 +2005 +1990 +2010 +2000 +2010 +2015 +1990 +1995 +2000 +2005 +2015 +2010 +Epoch (yr) +Epoch (year) +Epoch (year)6000 +HIPPARCOS +HIPPARCOS +0.90 +GAIA +GAIA +L +4000 +0 +8 +0.85 +μα* (mas/yr) +2000 +-1 +μs (mas/yr) +RV (m/s) +0.80 +-2 +0.75 +-2000 +12 +.4 +0.70 +-4000 +-5 +CORAVEL +-14 +HERMES +0.65 +-6000 +-6 +1000 +0.60 +O-C +O-C +0 +0 +-1000 +2 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A&A proofs: manuscript no. main +Fig. A.69: RV curve and proper motions of HD 210946 +Fig. A.70: RV curve and proper motions of HD 127392. We used a fixed RV offset of 195 m/s (Escorza et al. 2019b) +Mpri (M ) = 2.5+1.4 +1.0 +0.6 +0.9 +1.2 +1.5 +Msec (M ) +Msec (M ) = 0.86+0.26 +0.22 +3.0 +3.6 +4.2 +4.8 +a (AU) +a (AU) = 3.88+0.54 +0.53 +0.090 +0.105 +0.120 +0.135 +0.150 +e +e = 0.1094+0.011 +0.0089 +1.5 +3.0 +4.5 +6.0 +Mpri (M ) +60 +75 +90 +105 +120 +i ( ) +0.6 +0.9 +1.2 +1.5 +Msec (M ) +3.0 +3.6 +4.2 +4.8 +a (AU) +0.090 +0.105 +0.120 +0.135 +0.150 +e +60 +75 +90 +105 +120 +i ( ) +i ( ) = 113.9+4.7 +8.0 +Fig. A.71: Corner plot of HD 210946 +Mpri (M ) = 0.98+0.70 +0.53 +0.3 +0.6 +0.9 +1.2 +1.5 +Msec (M ) +Msec (M ) = 0.73+0.26 +0.24 +2.4 +3.0 +3.6 +4.2 +a (AU) +a (AU) = 3.08+0.49 +0.56 +0.050 +0.075 +0.100 +0.125 +0.150 +e +e = 0.088+0.017 +0.017 +0.8 +1.6 +2.4 +3.2 +Mpri (M ) +60 +75 +90 +105 +120 +i ( ) +0.3 +0.6 +0.9 +1.2 +1.5 +Msec (M ) +2.4 +3.0 +3.6 +4.2 +a (AU) +0.050 +0.075 +0.100 +0.125 +0.150 +e +60 +75 +90 +105 +120 +i ( ) +i ( ) = 119.2+4.7 +15 +Fig. A.72: Corner plot of HD 127392 +Article number, page 34 of 49 + +6000 +HIPPARCOS +HIPPARCOS +GAIA +GAIA +-2 +4000 +2000 +-2 E +1.2 +μα* (mas/yr) +μs (mas/yr) +RV (m/s) +-3 +-2000 +-6 +1.0 +-5 +-4000F +-6F +-8 +-6000F +0.8 +CORAVEL +-7E +HERMES +-8000 +SOPHIE +-8上 +1111 +2000 +2.5 [ +0.6 +2.5 +O-C +O-C +O-C +0 +0.0 +0.0 +-2.5E +-2.5E, +2015 +1990 +2000 +2010 +2020 +1990 +1995 +2000 +2005 +2010 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)HIPPARCOS +HIPPARCOS +0 +7500 +-20 + GAIA +GAIA +1.2 +5000 +-10F +2500 +μα* (mas/yr) +-30 + (mas/yr) +RV (m/s) +-20 +1.0 +0F +Mcomp(Mo) +35 +-2500上 +-30 +0.8 +-40 +-5000E +-40 +-7500 +45 +0.6 +CORAVEL +HERMES +-10000| +500 +10E +0.4 +O-C +O-C +0-0 +0 +-500 +V +-10E +2000 +1990 +2015 +1990 +1990 +1995 +2000 +2005 +2010 +2015 +2020 +1995 +2005 +2010 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. A.73: RV curve and proper motions of HD 143899. We used a fixed RV offset of 500 m/s (Jorissen et al. 2019). +Fig. A.74: RV curve and proper motions of HD 88562. We used a fixed RV offset of 500 m/s (Jorissen et al. 2019). +Mpri (M ) = 2.40+0.30 +0.30 +0.48 +0.56 +0.64 +0.72 +0.80 +Msec (M ) +Msec (M ) = 0.659+0.054 +0.055 +3.2 +3.4 +3.6 +3.8 +4.0 +a (AU) +a (AU) = 3.66+0.13 +0.15 +0.10 +0.15 +0.20 +0.25 +e +e = 0.181+0.035 +0.037 +1.6 +2.0 +2.4 +2.8 +3.2 +Mpri (M ) +116 +120 +124 +128 +i ( ) +0.48 +0.56 +0.64 +0.72 +0.80 +Msec (M ) +3.2 +3.4 +3.6 +3.8 +4.0 +a (AU) +0.10 +0.15 +0.20 +0.25 +e +116 +120 +124 +128 +i ( ) +i ( ) = 124.6+2.3 +2.5 +Fig. A.75: Corner plot of HD 143899 +Mpri (M ) = 0.99+0.30 +0.29 +0.30 +0.45 +0.60 +Msec (M ) +Msec (M ) = 0.477+0.080 +0.087 +2.0 +2.4 +2.8 +3.2 +a (AU) +a (AU) = 2.85+0.23 +0.27 +0.16 +0.18 +0.20 +0.22 +0.24 +e +e = 0.203+0.013 +0.013 +0.4 +0.8 +1.2 +1.6 +Mpri (M ) +75 +90 +105 +i ( ) +0.30 +0.45 +0.60 +Msec (M ) +2.0 +2.4 +2.8 +3.2 +a (AU) +0.16 +0.18 +0.20 +0.22 +0.24 +e +75 +90 +105 +i ( ) +i ( ) = 87.3+9.2 +9.2 +Fig. A.76: Corner plot of HD 88562 +Article number, page 35 of 49 + +6000 +HiPPARCOS +HIPPARCOs +GAIA +GAIA +34 +4000 +0.75 +32 +(uk/sew) * +2000 +μs (mas/yr) +RV (m/s) +2 +n +0.70 +30 +Mcomp(Mo) +1 F +*0n +28 +-2000 +oF +0.65 +26 +-4000 +1 +CORAVEL +0.60 +不 +HERMES +1000 +2.5 +2 +1 +O-C +O-C +0.55 +O-C +-1000 +-2.5 +2 +19901995 +1990 +20002005 +2010 20152020 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)10000 +HIPPARCOS +HIPPARCOS +GAIA +GAIA +0.65 +-21 +7500 E +1.0 +0.60 +5000E +(mas/yr) +-22 +μs (mas/yr) +0.55 +RV (m/s) +-2.0 +2500 +0.50 +-2.5 +-23 +*0n +-3.0 +0.45 +2500 +-24 +-3.5 +0.40 +-5000上 +CORAVEL +-4.0 +25 +HERMES +0.35 +-7500 +1000 +0.30 +O-C +0-0 +O-C +0 +1990 +2000 +2010 +2020 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A&A proofs: manuscript no. main +Fig. A.77: RV curve and proper motions of HD 221531 +Fig. A.78: RV curve and proper motions of HD 202400 +Mpri (M ) = 1.20+0.30 +0.29 +0.30 +0.45 +0.60 +0.75 +Msec (M ) +Msec (M ) = 0.582+0.084 +0.086 +2.4 +2.8 +3.2 +a (AU) +a (AU) = 2.97+0.20 +0.23 +0.152 +0.160 +0.168 +0.176 +e +e = 0.1653+0.0052 +0.0051 +0.4 +0.8 +1.2 +1.6 +2.0 +Mpri (M ) +60 +80 +100 +120 +i ( ) +0.30 +0.45 +0.60 +0.75 +Msec (M ) +2.4 +2.8 +3.2 +a (AU) +0.152 +0.160 +0.168 +0.176 +e +60 +80 +100 +120 +i ( ) +i ( ) = 58.5+2.8 +2.5 +Fig. A.79: Corner plot of HD 221531 +Mpri (M ) = 0.98+0.25 +0.25 +0.4 +0.6 +0.8 +1.0 +1.2 +Msec (M ) +Msec (M ) = 0.65+0.13 +0.12 +2.0 +2.4 +2.8 +3.2 +3.6 +a (AU) +a (AU) = 2.87+0.21 +0.24 +0.15 +0.30 +0.45 +e +e = 0.246+0.091 +0.090 +0.4 +0.8 +1.2 +1.6 +Mpri (M ) +30 +60 +90 +120 +150 +i ( ) +0.4 +0.6 +0.8 +1.0 +1.2 +Msec (M ) +2.0 +2.4 +2.8 +3.2 +3.6 +a (AU) +0.15 +0.30 +0.45 +e +30 +60 +90 +120 +150 +i ( ) +i ( ) = 61+67 +15 +Fig. A.80: Corner plot of HD 202400 +Article number, page 36 of 49 + +HIPPARCOS +HO +7500 E +35 +GAIA +-5 +GAIA +5000 +30 +0.8 +0 +-10 +2500 +K +(mas/yr) +μs (mas/yr) +RV (m/s) +25 +0E +-15 +0.7 +-2500 + *on +20 +-20 +-5000 E +15 +0.6 +-7500 +25 +CORAVEL +-10000 +10 +HERMES +30日 +0.5 +2500 +2.5 +5 +O-C +O-C +O-C +0.0 +0 +-2500 +-2.5 +1990 +1990 +2015 +1990 +1995 +2000 2005 +2010 +2015 +2020 +1995 +2000 +2005 +2010 +2015 +1995 +2000 +2005 +2010 +Epoch (yr) +Epoch (year) +Epoch (year)15000 +HIPPARCOS +GAIA +-25 +55 E +10000 +-30 +0.9 +μα* (mas/yr) +50 +(mas/yr) +RV (m/s) +5000 +-35 +0.8 +45 +0 +-40 +0.7 +40 +-45 +-5000 +CORAVEL +0.6 +1 +FEROS +35 +CORALIE +HIPPARCOS +SALT-HRS +-50 +GAIA +-10000 +0.5 +5000 +OTT +2.5 +O-C +O-C +0 +0.0 +0 +人工 +0 +0.4 +-5000 +-5 +-2.5 +1990 +1995 +2000 +2005 +2010 +2015 +1995 +2000 +2005 +2010 +2015 +2020 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year):':A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. A.81: RV curve and proper motions of HD 107574 +Fig. A.82: RV curve and proper motions of HD 58121 +Mpri (M ) = 1.11+0.15 +0.15 +0.6 +0.7 +0.8 +0.9 +Msec (M ) +Msec (M ) = 0.744+0.061 +0.061 +2.6 +2.8 +3.0 +3.2 +a (AU) +a (AU) = 2.99+0.11 +0.12 +0.072 +0.080 +0.088 +0.096 +e +e = 0.0832+0.0051 +0.0051 +0.8 +1.0 +1.2 +1.4 +1.6 +Mpri (M ) +165.0 +165.6 +166.2 +166.8 +167.4 +i ( ) +0.6 +0.7 +0.8 +0.9 +Msec (M ) +2.6 +2.8 +3.0 +3.2 +a (AU) +0.072 +0.080 +0.088 +0.096 +e +165.0 +165.6 +166.2 +166.8 +167.4 +i ( ) +i ( ) = 166.31+0.38 +0.39 +Fig. A.83: Corner plot of HD 107574 +Mpri (M ) = 2.9+1.4 +1.1 +0.25 +0.50 +0.75 +1.00 +1.25 +Msec (M ) +Msec (M ) = 0.67+0.19 +0.18 +2.4 +3.0 +3.6 +4.2 +a (AU) +a (AU) = 3.42+0.44 +0.48 +0.09 +0.12 +0.15 +0.18 +e +e = 0.135+0.019 +0.019 +1.5 +3.0 +4.5 +6.0 +7.5 +Mpri (M ) +60 +80 +100 +120 +i ( ) +0.25 +0.50 +0.75 +1.00 +1.25 +Msec (M ) +2.4 +3.0 +3.6 +4.2 +a (AU) +0.09 +0.12 +0.15 +0.18 +e +60 +80 +100 +120 +i ( ) +i ( ) = 120.5+3.7 +3.7 +Fig. A.84: Corner plot of HD 58121 +Article number, page 37 of 49 + +-135 +HIPPARCOS +HIPPARCOS +GAIA +GAIA +20 +2000F +0.85 +-140 +1000 +μα* (mas/yr) +μs (mas/yr) +15 +RV (m/s) +大天 +大 +0.80 +-145 +0 +10 +0.75 +-1000 +-150 +-2000 +5 +0.70 +-155 +CORAVEL +-3000 +不 +HERMES +0.65 +1000 +5 +O-C +O-C +O-C +0 +0 +0.60 +-1000 +5 +2010 +1990 +1990 +2000 +2020 +1990 +1995 +2000 +2005 +2010 +2015 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)6000 F +[DAO] +HIPPARCOS +HIPPARCOS + GAIA +GAIA +0 +4000F +1.0 +-1 +2000 +μα* (mas/yr) +μs (mas/yr) +0.9 +RV (m/s) +-2 +0 +0.8 +Mcomp(Mo) +口 +-3 +-1 +0.7 +-2000 +0.6 +-4000 +0.5 +2500 +1 F +0.4 +O-C +O-C +O-C +0 +中 +0F +-2500 +1 +2015 +1990 +2015 +1986 +1988 +1990 +1992 +1990 +1995 +2000 +2005 +2010 +1995 +2000 +2005 +2010 +Epoch (yr) +Epoch (year) +Epoch (year)A&A proofs: manuscript no. main +Fig. A.85: RV curve and proper motions of HD 31487 +Fig. A.86: RV curve and proper motions of HD 211594. We used a fixed RV offset of 500 m/s (Jorissen et al. 2019). +Mpri (M ) = 2.54+0.60 +0.59 +1.2 +1.5 +1.8 +2.1 +Msec (M ) +Msec (M ) = 1.59+0.22 +0.22 +2.7 +3.0 +3.3 +3.6 +3.9 +a (AU) +a (AU) = 3.27+0.20 +0.23 +0.025 +0.050 +0.075 +0.100 +e +e = 0.037+0.018 +0.018 +1.6 +2.4 +3.2 +4.0 +4.8 +Mpri (M ) +50 +75 +100 +125 +i ( ) +1.2 +1.5 +1.8 +2.1 +Msec (M ) +2.7 +3.0 +3.3 +3.6 +3.9 +a (AU) +0.025 +0.050 +0.075 +0.100 +e +50 +75 +100 +125 +i ( ) +i ( ) = 32.1+1.7 +1.4 +Fig. A.87: Corner plot of HD 31487 +Mpri (M ) = 2.11+0.84 +0.76 +0.25 +0.50 +0.75 +1.00 +Msec (M ) +Msec (M ) = 0.55+0.16 +0.14 +2.0 +2.4 +2.8 +3.2 +a (AU) +a (AU) = 2.75+0.30 +0.35 +0.02 +0.04 +0.06 +0.08 +0.10 +e +e = 0.058+0.013 +0.013 +1 +2 +3 +4 +Mpri (M ) +50 +75 +100 +125 +i ( ) +0.25 +0.50 +0.75 +1.00 +Msec (M ) +2.0 +2.4 +2.8 +3.2 +a (AU) +0.02 +0.04 +0.06 +0.08 +0.10 +e +50 +75 +100 +125 +i ( ) +i ( ) = 123+11 +16 +Fig. A.88: Corner plot of HD 211594 +Article number, page 38 of 49 + +HIPPARCOS +HIPPARCOS +6000 + GAIA +GAIA +2.0 +4000 +0 +μα* (mas/yr) +0 +2000 +(mas/yr) +(s/u) +1.8 +-5 +0 +-5 A +-2000 +-10F +(Mo) +4000 +-10 +6000F +1.4 +-15 +DAO +8000 +HERMES +-15 +2000F1 +1.2 +5A +0-0 +0氏 +0F +0 +0 +-5 +-2000E +2000 +1980 +1990 +2010 +2020 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)HIPPARCOS +6000 +CORAVEL +HIPPARCOS +24 +16F +HERMES + GAIA +GAIA +4000 +0.9 +14上 +22 +μα* (mas/yr) +2000 +12 +(μ/sew) 9r +0.8 +RV (m/s) +20 +10 +oF +0.7 +18 +8 +-2000 +0.6 +16 +6 +-4000H +0.5 +14 +4 +-6000E +0.4 +1000 +2.5 +2.5 +O-C +O-C +O-C +0.0 +0.3 +0 +0.0 +-2.5 +2.5 +-1000 +1990 +1995 +20002005 + 20102015 +1990 +1995 +20002005 +1990 +2000 +2010 +2020 + 20102015 +Epoch (yr) +Epoch (year) +Epoch (year)A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. A.89: RV curve and proper motions of HD 34654 +Fig. A.90: RV curve and proper motions of HD 92626 +Mpri (M ) = 1.20+0.20 +0.20 +0.5 +0.6 +0.7 +0.8 +Msec (M ) +Msec (M ) = 0.641+0.061 +0.063 +2.10 +2.25 +2.40 +2.55 +a (AU) +a (AU) = 2.36+0.10 +0.12 +0.1075 +0.1100 +0.1125 +0.1150 +e +e = 0.1114+0.0016 +0.0016 +0.75 +1.00 +1.25 +1.50 +1.75 +Mpri (M ) +64 +72 +80 +88 +i ( ) +0.5 +0.6 +0.7 +0.8 +Msec (M ) +2.10 +2.25 +2.40 +2.55 +a (AU) +0.1075 +0.1100 +0.1125 +0.1150 +e +64 +72 +80 +88 +i ( ) +i ( ) = 74.3+5.2 +4.3 +Fig. A.91: Corner plot of HD 34654 +Mpri (M ) = 3.2+1.7 +1.4 +0.6 +0.9 +1.2 +1.5 +Msec (M ) +Msec (M ) = 0.90+0.27 +0.27 +2.0 +2.5 +3.0 +3.5 +4.0 +a (AU) +a (AU) = 2.96+0.42 +0.48 +0.015 +0.030 +0.045 +0.060 +e +e = 0.014+0.014 +0.010 +2 +4 +6 +8 +Mpri (M ) +60 +75 +90 +105 +120 +i ( ) +0.6 +0.9 +1.2 +1.5 +Msec (M ) +2.0 +2.5 +3.0 +3.5 +4.0 +a (AU) +0.015 +0.030 +0.045 +0.060 +e +60 +75 +90 +105 +120 +i ( ) +i ( ) = 84.7+8.8 +8.8 +Fig. A.92: Corner plot of HD 92626 +Article number, page 39 of 49 + +HERMES +不 +HIPPARCOS +10000E +HIPPARCOS +A +GAIA +GAIA +7500E +200 +0.75 +-115 +A +5000 +(uk/sew) +-120 +(uk/sew) +RV (m/s) +180 +0.70 +2500 +-125 +oF +*"n +160 +-130 +0.65 +-2500 +4 +-135 +-5000E +140 +0.60 +-140E +-7500氏 +-145 +120 +0.55 +250F +25 +10 +O-C +T +O-C +O-C +0 +0 +0.50 +-10 +-250 +25 +2012 +2014 +2016 +2018 +2020 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)CORAVEL +HIPPARCOS +HIPPARCOS +-42 +7500 +GAIA +GAIA +1.4 +-44H +5000E +4 +(mas/yr) +2500 +(mas/yr) +(s/w) +-46 +1.2 +2 +o +RV( +-48 +*on +1.0 +-2500F +0 +-50 +-5000 +0.8 +2 +-7500 +-52 +0.6 +500 +2.5E +O-C +0 +0.0 +-500 +2.5 +2005 +2015 +1990 +1995 +2005 +2010 +2015 +1986 +1988 +1990 +1992 +1994 +2000 +1990 +1995 +2000 +2010 +Epoch (yr) +Epoch (year) +Epoch (year)A&A proofs: manuscript no. main +Fig. A.93: RV curve and proper motions of HD 50264. +Fig. A.94: RV curve and proper motions of HD 49841 +Mpri (M ) = 0.90+0.30 +0.29 +0.4 +0.6 +0.8 +1.0 +Msec (M ) +Msec (M ) = 0.63+0.13 +0.13 +1.5 +1.8 +2.1 +2.4 +a (AU) +a (AU) = 2.12+0.17 +0.21 +0.04 +0.08 +0.12 +0.16 +e +e = 0.077+0.018 +0.016 +0.4 +0.8 +1.2 +1.6 +Mpri (M ) +60 +80 +100 +120 +i ( ) +0.4 +0.6 +0.8 +1.0 +Msec (M ) +1.5 +1.8 +2.1 +2.4 +a (AU) +0.04 +0.08 +0.12 +0.16 +e +60 +80 +100 +120 +i ( ) +i ( ) = 104+13 +40 +Fig. A.95: Corner plot of HD 50264 +Mpri (M ) = 2.85+0.29 +0.30 +1.0 +1.5 +2.0 +2.5 +Msec (M ) +Msec (M ) = 0.819+0.16 +0.079 +2.6 +2.8 +3.0 +3.2 +a (AU) +a (AU) = 2.82+0.10 +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +e +e = 0.162+0.015 +0.015 +2.0 +2.4 +2.8 +3.2 +3.6 +Mpri (M ) +30 +60 +90 +120 +150 +i ( ) +1.0 +1.5 +2.0 +2.5 +Msec (M ) +2.6 +2.8 +3.0 +3.2 +a (AU) +0.12 +0.14 +0.16 +0.18 +0.20 +e +30 +60 +90 +120 +150 +i ( ) +i ( ) = 109+15 +19 +Fig. A.96: Corner plot of HD 49841 +Article number, page 40 of 49 + +150 +10000 +HIPPARCOS +HIPPARCOS +70 E +GAIA +GAIA +140 +0.9 +60 +5000F +130 +μα* (mas/yr) +50 +(uK/sew) 9r +0.8 +RV (m/s) +120 +o +40 +110 +Mcomp(Mo) +0.7 +30 +100 +-5000 +20 +0.6 +90 + CORAVEL +10 +HERMES +0.5 +-10000 +80 + SALT-HRS +oE +25 F +1000F +0.4 +10E +O-C +O-C +O-C +0 +0 +0 +一 +-1000F +-10 +1990 +1995 + 2000　2005　2010　2015 +2020 +1990 +1995 +200020052010 +2015 +1990 +20002005 +1995 + 2010　2015 +Epoch (yr) +Epoch (year) +Epoch (year)HIPPARCOS +HIPPARCOS + CORAVEL +GAIA +8000F + GAIA +1.6 +6000 +-2 +4000 +(μK/sew) *r +2 +(mas/yr) +RV (m/s) +-4 +1.4 +2000上 +0 +-6 +oE +1.2 +-2000E +-2 +-8 +-4000 +1.0 +-10H +-6000 +2.5 F +0.8 +1000 +2.5 +O-C +0 +0.0 +0.0 +-1000 +-2.5 +-2.5 LM +1986 +1988 +1990 +1992 +1994 +1990 +1995 +2000 +2005 + 2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. A.97: RV curve and proper motions of HD 58368 +Fig. A.98: RV curve and proper motions of HD 207585 +Mpri (M ) = 2.57+0.61 +0.59 +0.8 +1.6 +2.4 +3.2 +Msec (M ) +Msec (M ) = 0.66+0.17 +0.11 +1.5 +1.8 +2.1 +2.4 +2.7 +a (AU) +a (AU) = 2.23+0.16 +0.17 +0.16 +0.20 +0.24 +0.28 +e +e = 0.217+0.024 +0.023 +0.8 +1.6 +2.4 +3.2 +4.0 +Mpri (M ) +30 +60 +90 +120 +150 +i ( ) +0.8 +1.6 +2.4 +3.2 +Msec (M ) +1.5 +1.8 +2.1 +2.4 +2.7 +a (AU) +0.16 +0.20 +0.24 +0.28 +e +30 +60 +90 +120 +150 +i ( ) +i ( ) = 78+27 +25 +Fig. A.99: Corner plot of HD 58368 +Mpri (M ) = 0.91+0.30 +0.28 +0.50 +0.75 +1.00 +1.25 +1.50 +Msec (M ) +Msec (M ) = 0.57+0.12 +0.11 +1.4 +1.6 +1.8 +2.0 +a (AU) +a (AU) = 1.72+0.14 +0.16 +0.04 +0.08 +0.12 +0.16 +e +e = 0.031+0.033 +0.022 +0.4 +0.8 +1.2 +1.6 +Mpri (M ) +50 +75 +100 +125 +i ( ) +0.50 +0.75 +1.00 +1.25 +1.50 +Msec (M ) +1.4 +1.6 +1.8 +2.0 +a (AU) +0.04 +0.08 +0.12 +0.16 +e +50 +75 +100 +125 +i ( ) +i ( ) = 93+18 +20 +Fig. A.100: Corner plot of HD 207585 +Article number, page 41 of 49 + +DAO +8上 +HIPPARCOS +8E +HIPPARCOS +GAIA +GAIA +8000 E +6E +6 +1.6 +站 +6000F +4 +4 +4000E +μα* (mas/yr) +1.4 +(mas/yr) +2 +(s/wu) +2 +2000 +1.2 +Mcomp(Mo) +0 +o F +-2 +-2000 +-2 A +1.0 +一 +4000 +-4 +0.8 +6F +-6000F +-6 +-8 +0.6 +1000 +O-C +O-C +0 +0E +0 +0 +0.4 +-1 +1980 +1982 +1984 +1986 +1988 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)HIPPARCOS +HIPPARCOS +GAIA + GAIA +10000 +-20F +30F +0.9 +5000 +μα* (mas/yr) +-30 +(mas/yr) +(s/u) +20 +0.8 +0 +-40 +10 +-5000 +-50 +OF +CORAVEL +-10000 +HERMES + SALT-HRS +0.5 +-60 +2500 F +10E +2.5 +O-C +0.4 +0F +OF +0.0 +0 +-10 +-2.5 +2500 E +1990 1995 2000 2005 + 2010 2015 2020 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 + 20102015 +Epoch (yr) +Epoch (year) +Epoch (year)A&A proofs: manuscript no. main +Fig. A.101: RV curve and proper motions of HD 44896 +Fig. A.102: RV curve and proper motions of HD 199939 +Mpri (M ) = 2.99+0.40 +0.40 +0.8 +1.2 +1.6 +2.0 +2.4 +Msec (M ) +Msec (M ) = 1.00+0.22 +0.12 +1.95 +2.10 +2.25 +2.40 +2.55 +a (AU) +a (AU) = 2.288+0.094 +0.10 +0.015 +0.030 +0.045 +0.060 +e +e = 0.019+0.013 +0.012 +1.8 +2.4 +3.0 +3.6 +4.2 +Mpri (M ) +50 +75 +100 +125 +150 +i ( ) +0.8 +1.2 +1.6 +2.0 +2.4 +Msec (M ) +1.95 +2.10 +2.25 +2.40 +2.55 +a (AU) +0.015 +0.030 +0.045 +0.060 +e +50 +75 +100 +125 +150 +i ( ) +i ( ) = 78+35 +21 +Fig. A.103: Corner plot of HD 44896 +Mpri (M ) = 2.97+0.89 +0.67 +0.6 +0.8 +1.0 +1.2 +Msec (M ) +Msec (M ) = 0.73+0.14 +0.11 +1.75 +2.00 +2.25 +2.50 +a (AU) +a (AU) = 2.12+0.18 +0.16 +0.26 +0.28 +0.30 +0.32 +e +e = 0.281+0.013 +0.012 +2 +3 +4 +5 +6 +Mpri (M ) +60 +80 +100 +120 +i ( ) +0.6 +0.8 +1.0 +1.2 +Msec (M ) +1.75 +2.00 +2.25 +2.50 +a (AU) +0.26 +0.28 +0.30 +0.32 +e +60 +80 +100 +120 +i ( ) +i ( ) = 82+21 +13 +Fig. A.104: Corner plot of HD 199939 +Article number, page 42 of 49 + +10000 E +8F +HIPPARCOS +HIPPARCOS +GAIA +GAIA +7500E +2.0 +5000 +1.8 +-6 +μα* (mas/yr) +2500E +(mas/yr) +RV (m/s) +2 +1.6 +-8 +0 +-2500 E +-10 +-2 +1.2 +-5000 +-12 +-7500 E +-4F +1.0 +-14F +-6 +-10000 +CORAVEL +0.8 +5 E +500 +0.6 +O-C +0 +0一 +-500F +-5E. +-21 +1986 +1988 +1990 +1992 +1994 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)HIPPARCOS +10000 +HIPPARCOS +-16 +GAIA +GAIA +1.1 +7500E +-18 +0 +1.0 +5000 +(mas/yr) +(μk/sew) +RV (m/s) +-2 +-20 +2500 +0.9 +-22 +-4 +*n +-2500 +-24 +0.7 +-5000 +DAO +-8上 +HERMES +-26 +SOPHIE +0.6 +-7500E +L +2000F +2 +2.5 +O-C +0.5 +O-C +C +0 +0.0 +0 +TMMMMNNMMMMMM +-2.5 +-2000 E +-2E +2000 +1980 +1990 +2010 +2020 +1990 +1995 +20002005 + 2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. A.105: RV curve and proper motions of HD 123585 +Fig. A.106: RV curve and proper motions of HD 24035 +Mpri (M ) = 1.12+0.29 +0.29 +0.4 +0.6 +0.8 +1.0 +Msec (M ) +Msec (M ) = 0.65+0.11 +0.10 +1.05 +1.20 +1.35 +1.50 +1.65 +a (AU) +a (AU) = 1.41+0.10 +0.11 +0.05 +0.10 +0.15 +0.20 +e +e = 0.025+0.038 +0.018 +0.4 +0.8 +1.2 +1.6 +2.0 +Mpri (M ) +60 +80 +100 +120 +i ( ) +0.4 +0.6 +0.8 +1.0 +Msec (M ) +1.05 +1.20 +1.35 +1.50 +1.65 +a (AU) +0.05 +0.10 +0.15 +0.20 +e +60 +80 +100 +120 +i ( ) +i ( ) = 90+19 +19 +Fig. A.107: Corner plot of HD 123585 +Mpri (M ) = 1.82+0.68 +0.58 +0.6 +1.2 +1.8 +2.4 +3.0 +Msec (M ) +Msec (M ) = 0.76+0.25 +0.17 +1.0 +1.2 +1.4 +1.6 +1.8 +a (AU) +a (AU) = 1.41+0.14 +0.15 +0.02 +0.04 +0.06 +0.08 +e +e = 0.014+0.016 +0.010 +1 +2 +3 +4 +Mpri (M ) +30 +60 +90 +120 +150 +i ( ) +0.6 +1.2 +1.8 +2.4 +3.0 +Msec (M ) +1.0 +1.2 +1.4 +1.6 +1.8 +a (AU) +0.02 +0.04 +0.06 +0.08 +e +30 +60 +90 +120 +150 +i ( ) +i ( ) = 71+31 +22 +Fig. A.108: Corner plot of HD 24035 +Article number, page 43 of 49 + +HIPPARCOS +HIPPARCOS +-10 +GAIA +GAIA +50 +10000 +0.9 +-15 +45 +-20 +5000 +μα* (mas/yr) +0.8 + (mas/yr) +40 +RV (m/s) +-25 +35 +-30 +0.7 +Mcomp(Mo) +oF +30 +-35 +0.6 +-5000 F +25 E +-40 +0.5 +20 E +-45F +-10000 +ΦCORAVEL +0.4 +5 +10日 +1000 +O-C +OF +0 +0 +0 +0 +0.3 +-1000 +-5 +1990 +1995 +2000 +20102015 +1990 +1995 +2000 +2005 +1991 +1992 +1993 +1994 +1995 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)HIPPARCOS +HIPPARCOS +10000 + GAIA +GAIA +1.8 +45 +0 +-5 +40 +1.6 +5000F +(u/sew) +(mas/yr) +-10 +(s/w) +35 +1.4 +of +-15 +30 +1.2 +*on +-20 +-5000 +25 +1.00 +25 +20 +—10000 +0.8 +CORAVEL +-30 +500F +2 +0.6 +O-C +O-C +O-C +0 +0 +-500M +-5 +-2 +1986 +1988 +1990 +1992 +1994 +1990 +1995 +2000 +2005 +2010 +2015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A&A proofs: manuscript no. main +Fig. A.109: RV curve and proper motions of HD 224621 +Fig. A.110: RV curve and proper motions of HD 87080. We used a fixed RV offset of 248 m/s (Escorza et al. 2019b). +Mpri (M ) = 0.85+0.10 +0.10 +0.4 +0.8 +1.2 +1.6 +2.0 +Msec (M ) +Msec (M ) = 0.66+0.26 +0.13 +0.9 +1.0 +1.1 +1.2 +1.3 +a (AU) +a (AU) = 1.029+0.061 +0.048 +0.015 +0.030 +0.045 +e +e = 0.020+0.012 +0.012 +0.60 +0.75 +0.90 +1.05 +Mpri (M ) +30 +60 +90 +120 +150 +i ( ) +0.4 +0.8 +1.2 +1.6 +2.0 +Msec (M ) +0.9 +1.0 +1.1 +1.2 +1.3 +a (AU) +0.015 +0.030 +0.045 +e +30 +60 +90 +120 +150 +i ( ) +i ( ) = 31.0+6.0 +6.7 +Fig. A.111: Corner plot of HD 224621 +Mpri (M ) = 1.36+0.42 +0.41 +0.6 +0.9 +1.2 +1.5 +Msec (M ) +Msec (M ) = 0.70+0.18 +0.14 +0.90 +1.05 +1.20 +a (AU) +a (AU) = 1.054+0.089 +0.10 +0.100 +0.125 +0.150 +0.175 +0.200 +e +e = 0.162+0.016 +0.016 +0.5 +1.0 +1.5 +2.0 +2.5 +Mpri (M ) +50 +75 +100 +125 +150 +i ( ) +0.6 +0.9 +1.2 +1.5 +Msec (M ) +0.90 +1.05 +1.20 +a (AU) +0.100 +0.125 +0.150 +0.175 +0.200 +e +50 +75 +100 +125 +150 +i ( ) +i ( ) = 60+11 +12 +Fig. A.112: Corner plot of HD 87080 +Article number, page 44 of 49 + +10000 +20 +HIPPARCOS +HIPPARCOS +GAIA +GAIA +7500 +1.1 +-80 +0 +5000 +1.0 +-100 +(mas/yr) +(uk/sew) +2500 +(s/u) +-20 +0.9 +-120 +0 +0.8 W +-40 +comp(M。) +-2500 +-140 +0.7 +-5000 +-60 +0.6 +LB91 +-160 +亚 +-7500 +CES +CORAVEL +-80 +0.5 +25 +10 +1000 +0.4 +-0 +T +O-C +0 +0 +-1000 +-25 +-10 +1990 +1980 +1985 +1995 +1990 +1995 +2000200520102015 +1990 +199520002005　2010　2015 +Epoch (yr) +Epoch (year) +Epoch (year)90 +HIPPARCOS +HIPPARCOS +GAIA + GAIA +10000 +-50 +1.6 +-55 +80 +5000 +μα* (mas/yr) +1.4 +(u/sew) +-60 +RV (m/s) +70 +0 +-65 +1.2 +Mcomp(Mo) +-70 +-5000 +60 +1.0 +-75 +-10000H +0.8 +CORAVEL +50 +-80 +HERMES +-15000 [ +0.6 +500 +5 +O-C +0-0 +0-0 +0 +0 +0.4 +500 +-5 +2010 +20102015 +1990 +1995 +2000 +2005 +2015 +1990 +1995 +2000 +2005 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. A.113: RV curve and proper motions of HD 121447 +Fig. A.114: RV curve and proper motions of HD 77247 +Mpri (M ) = 1.59+0.29 +0.29 +0.6 +1.2 +1.8 +2.4 +3.0 +Msec (M ) +Msec (M ) = 0.72+0.36 +0.19 +0.7 +0.8 +0.9 +1.0 +a (AU) +a (AU) = 0.848+0.054 +0.052 +0.015 +0.030 +0.045 +e +e = 0.0121+0.010 +0.0081 +0.8 +1.2 +1.6 +2.0 +2.4 +Mpri (M ) +30 +60 +90 +120 +150 +i ( ) +0.6 +1.2 +1.8 +2.4 +3.0 +Msec (M ) +0.7 +0.8 +0.9 +1.0 +a (AU) +0.015 +0.030 +0.045 +e +30 +60 +90 +120 +150 +i ( ) +i ( ) = 59+76 +24 +Fig. A.115: Corner plot of HD 121447 +Mpri (M ) = 4.61+0.56 +0.65 +0.45 +0.60 +0.75 +0.90 +1.05 +Msec (M ) +Msec (M ) = 0.541+0.11 +0.059 +0.56 +0.60 +0.64 +0.68 +a (AU) +a (AU) = 0.632+0.024 +0.032 +0.088 +0.096 +0.104 +0.112 +0.120 +e +e = 0.1080+0.0042 +0.0049 +3.2 +4.0 +4.8 +5.6 +6.4 +Mpri (M ) +50 +75 +100 +125 +i ( ) +0.45 +0.60 +0.75 +0.90 +1.05 +Msec (M ) +0.56 +0.60 +0.64 +0.68 +a (AU) +0.088 +0.096 +0.104 +0.112 +0.120 +e +50 +75 +100 +125 +i ( ) +i ( ) = 98+25 +22 +Fig. A.116: Corner plot of HD 77247 +Article number, page 45 of 49 + +22.5 +HIPPARCOS +HIPPARCOS +-32.5 + GAIA +GAIA +10000F +20.0 +2.0 +-35.0 +17.5 +1.8 +5000 +-37.5 +(uk/sew) +15.0 +(uk/sew) +(s/w) +1.6 +-40.0 +12.5 +of +-42.5 +1.4W +10.0 +-5000上: +7.5 +1.2 +-47.5 +5.0 +1.0 +-10000 +-50.0[ +2.5 +0.8 +CORAVEL +2.5 +1000 +0.6 +oF +0.0 +0 +0 +-1000 +-2.5 +6 +1986 +199019952000 2005 20102015 +19901995 20002005 2010 2015 +1988 +1990 +1992 +1994 +Epoch (yr) +Epoch (year) +Epoch (year)8F +HIPPARCOS +HIPPARCOS +-4 +10000 +GAIA +GAIA +6 +7500 +-6 +0.9 +5000 +-8 +μα* (mas/yr) +(s/w) +2500 +0.8 +-10 +2 +0 +-12 +0 +0.7 +-2500 +-14 +-5000 +-2 +-16 +0.6 +-75008 +-18 +-1988 +0.5 +5 +2.5 +O-C +O-C +O-C +0.0 +0 +0.4 +2.5 +2500 +1980 +1982 +1984 +2010 2011 +1990 +1995 +2000 +200520102015 +1990 +1995 +2000 +2005 +2010 +2015 +Epoch (yr) +Epoch (year) +Epoch (year)A&A proofs: manuscript no. main +Appendix B: Corner plots of HD 218356 +Article number, page 46 of 49 + +A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Mpri (M ) = 5.1+2.2 +1.4 +0.15 +0.30 +0.45 +Msec (M ) +Msec (M ) = 0.128+0.063 +0.033 +0.7 +0.8 +0.9 +1.0 +a (AU) +a (AU) = 0.785+0.10 +0.079 +0.05 +0.10 +0.15 +0.20 +e +e = 0.072+0.048 +0.045 +5.0 +7.5 +10.0 +12.5 +Mpri (M ) +40 +80 +120 +160 +i ( ) +0.15 +0.30 +0.45 +Msec (M ) +0.7 +0.8 +0.9 +1.0 +a (AU) +0.05 +0.10 +0.15 +0.20 +e +40 +80 +120 +160 +i ( ) +i ( ) = 90+42 +41 +Mpri (M ) = 5.1+2.2 +1.4 +0.6 +0.9 +1.2 +1.5 +Msec (M ) +Msec (M ) = 0.85+0.25 +0.18 +18 +24 +30 +36 +a (AU) +a (AU) = 22.1+3.6 +2.8 +0.2 +0.4 +0.6 +0.8 +e +e = 0.39+0.13 +0.12 +5.0 +7.5 +10.0 +12.5 +Mpri (M ) +120 +135 +150 +165 +i ( ) +0.6 +0.9 +1.2 +1.5 +Msec (M ) +18 +24 +30 +36 +a (AU) +0.2 +0.4 +0.6 +0.8 +e +120 +135 +150 +165 +i ( ) +i ( ) = 157.2+4.3 +5.1 +Fig. B.1: Corner plots of the inner (left) and outer (right) orbit of HD 218356. +Article number, page 47 of 49 + +A&A proofs: manuscript no. main +Appendix C: Two possible fits for HD 201657 +Article number, page 48 of 49 + +A. Escorza and R. J. De Rosa: Barium and related stars, and their white-dwarf companions III. The masses of the white dwarfs +Fig. C.1: Possible orbit for HD 201657 with a smaller eccentricity. Compatible with the orbit published by Jorissen et al. (2019). +Fig. C.2: Possible orbit for HD 201657 with a larger eccentricity. Twice the period published by Jorissen et al. (2019). +Fig. C.3: Corner plots associated with the fits for HD 201657 shown in figure C.1 (left) and C.2 (right). +Article number, page 49 of 49 + +HIPPARCOS +36 +GAIA +4000 +12 +1.0 +34 E +2000 +μα* (mas/yr) +0.9 +10 +μs (mas/yr) +RV (m/s) +30 +0.8 +0 +Mcomp(Mo) +8 +28 +0.7 +-2000 +6 +26 +0.6 +4 +-4000F +CORAVEL +HIPPARCOS +24 +HERMES +GAIA +0.5 +2.5 +1000 +O-C +0.4 +O-C +O-C +0 +0.0 +AA +0 +-1000 +-1 +-2.5E +1980 +1990 +2000 +2010 +2020 +1990 +1995 +1990 +1995 +2000 +2005 +2010 + 2015 +Epoch (yr) +Epoch (year) +Epoch (year)10000 +30 F +HIPPARCOS +HIPPARCOS +55 + GAIA +O GAIA +8000 +2.00 +50 [ +20 +6000 +45 +1.75 +(uk/sew) +(s/u) +4000 +10F +1.50 +2000 +1.25 +-2000 +1.00 +25 +CORAVEL +-10 +-4000F +不 +HERMES +0.75 +2000F +O-C +O-C +0.50 +-2000 +1980 +1990 +2000 +2010 +2020 +1990 +1995 +2000 +2005 +20102015 +1990 +1995 +2000 +2005 +20102015 +Epoch (yr) +Epoch (year) +Epoch (year)Mpr (Mo) = 2.00+0:81 +Mpri(Mo) = 1.81±0:95 +Msec (Mo) = 0.66±0:19 +? +2.4 +(Mo) +Msec (Mo) +0.8 +6 +4 +a (AU) = 3.878:3 +a (AU) = 6.23+1:81 +(AU) +(nv) e += 0.159±0:06 +0.72-0:1 +LODO +1000 +0.32 +i()= 152.7: +i(°) = 32.6 +160 +75 +C +心 +0.90 +4.5 +Mpri (Mo) +Msec(Mo) +a (AU) +e +Mpri (Mo) +Msec(Mo) +a (AU) +i(°) +e +i(°) \ No newline at end of file diff --git a/EdE2T4oBgHgl3EQf9wni/content/tmp_files/load_file.txt b/EdE2T4oBgHgl3EQf9wni/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..05c266f5e9d1cf4f16d65059ea2ea93cd8d0af91 --- /dev/null +++ b/EdE2T4oBgHgl3EQf9wni/content/tmp_files/load_file.txt @@ -0,0 +1,5942 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf,len=5941 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main ©ESO 2023 January 12, 2023 Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza1 and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa1 European Southern Observatory, Alonso de Córdova 3107, Vitacura, Santiago, Chile e-mail: ana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='escorza@eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='org January 12, 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Masses are one of the most difficult stellar properties to measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In the case of the white-dwarf (WD) companions of Barium (Ba) stars, the situation is worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' These stars are dim, cool, and difficult to observe via direct methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' However, Ba stars were polluted by the Asymptotic Giant Branch (AGB) progenitors of these WDs with matter rich in heavy elements, and the properties of their WD companions contain key information about binary interaction processes involving AGB stars and about the slow-neutron-capture(s)- process of nucleosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' With this study, we aim to determine accurate and assumption-free masses for the WD companions of as many Ba stars as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We want to provide new observational constraints that can help us learn about the formation and evolution of these post- interaction binary systems and about the nucleosythesis processes that took place in the interiors of their AGB progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We combined archival radial-velocity data with Hipparcos and Gaia astrometry using the software package orvara, a code designed to simultaneously fit a single Keplerian model to any combination of these types of data using a parallel-tempering Markov chain Monte Carlo method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We adopted Gaussian priors for the Ba star masses and for the parallaxes, and assumed uninformative priors for the orbital elements and the WD masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We determined new orbital inclinations and companion masses for 60 Ba star systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' These results include a couple of new orbits and several improved orbits for the longest-period systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Additionally, we unravelled a new triple system that was not known before and constrained the orbits and the masses of the two companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The WD mass distribution presented in this work is compatible with that of field WDs and with the distributions published before for Ba star companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A few WD companions have masses higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 M⊙, considering 1-σ uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This indicates that they might come from AGB stars that are more massive than 3 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' These masses are higher than what the abundance ratios on Ba star atmospheres and theoretical models of the s-process of nucleosynthesis seem to expect, raising interesting questions about the formation of these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' white dwarfs - stars: late-type - stars: chemically peculiar - binaries: spectroscopic - astrometry - stars: evolution 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Introduction About half of the elements heavier than iron are synthesized by the slow neutron capture (s-) process of nucleosynthesis (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Burbidge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1957;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Clayton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1961;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Käppeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The main astrophysical site that meets the appropriate condi- tions for the s-process to operate is the helium-rich intershell in the interiors of thermally pulsing Asymptotic Giant Branch (tp-AGB) stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Lugaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2003b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Cristallo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Karakas 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Käppeler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' However, the overabun- dance of s-process elements on the surface of a star is not a unique feature of AGB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Barium (Ba) stars are an example of s-process enriched objects that have not reached the tp-AGB phase yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' They are known to form when an AGB companion pollutes them in a binary system (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' McClure et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Mc- Clure 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Udry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1998a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The mass donors in these systems evolved off the AGB long ago and are now dim white dwarfs (WD), while the accretors – the Ba stars – are observed on the main sequence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' North & Duquennoy 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Jorissen & Boffin 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' North et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1994, 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Pereira 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019b), the red-giant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Bidelman & Keenan 1951;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' McClure 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Udry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1998b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Jorissen 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019), and the AGB (as extrinsic S stars, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1998, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Shetye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2020) phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Although their exact formation channel and the mass- transfer mechanisms involved are not well understood (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Tout & Eggleton 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Soker 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Pols et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Bonaˇci´c Marinovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Izzard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Dermine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Abate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Saladino & Pols 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2023), our knowledge about the spectroscopic orbital parameters of Ba star systems and about the stellar properties of the Ba stars them- selves is generally well established (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Additionally, the evolutionary link between dwarf and giant Ba stars is well ac- cepted (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' However, not much is known about the WD companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The mass-function distribution of Ba star systems is consistent with a narrow distribution of com- panion masses peaking at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 M⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Webbink 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' McClure & Woodsworth 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Merle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019a), but very few abso- lute masses have been determined, since there is normally no in- formation about the orbital inclinations of these systems (a few exceptional cases were published by Pourbaix & Jorissen 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019, among others, by com- bining the orbital parameters of Ba stars with Hipparcos astro- Article number, page 1 of 49 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04232v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='SR] 10 Jan 2023 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main metric data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' These WDs are cool, dim, and directly undetectable in most cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' although, Böhm-Vitense et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (1984, 2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Gray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2011), among others detected UV excess flux attributable to the WD in a few Ba star systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the WD companions of Ba stars contain im- portant information about the AGB progenitors and the nucle- osynthesis processes that took place in their interiors, and they are important input for binary interaction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Even though mixing and dilution processes such as thermohaline mixing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Proffitt & Michaud 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Charbonnel & Zahn 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Stancliffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Stancliffe & Glebbeek 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Aoki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2008), ro- tationally induced mixing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Denissenkov & Tout 2000), or atomic diffusion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Matrozis & Stancliffe 2016, 2017) might impact the final level of s-process abundance on Ba stars, corre- lations between these abundances and the WD mass can give us observational information about the efficiency of the s-process at different masses and metallicities and help us constrain AGB models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Cseh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2022) and mass-transfer and dilution models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Stancliffe 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The ratio between the amount of heavy s-process elements (hs), such as Ba, La, or Ce, and light s-process elements (ls), such as Sr, Y, or Zr, on the surface of Ba stars suggests that the material accreted by these stars was syn- thesized by low-mass AGB stars (< 3 M⊙ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Lugaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2003a, 2012, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Cseh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Karinkuzhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2018), which still needs to be confirmed by measuring these WD masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Ad- ditionally, Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019) suggested that WD compan- ions of strong Ba giants (based on the Ba index introduced by Warner 1965) are more massive on average than the WD com- panions of mild Ba stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' However, most of their masses were determined under the assumption of a constant (or very narrow distribution of) Q = M3 WD/(MBa + MWD)2 as proposed by Web- bink (1988) and McClure & Woodsworth (1990), so this trend still needs to be confirmed with assumption-free measurements of WD masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In the first two papers of this series, Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019) and Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b) collected old and new radial-velocity (RV) data to study the orbits of giant and dwarf Ba stars, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Additionally, we used spectroscopically-determined stellar parameters and Gaia DR2 distances (Lindegren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2018) to locate these stars on the Hertzsprung–Russell diagram (HRD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' By comparing their loca- tion on the HRD with STAREVOL evolutionary tracks (Siess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Siess 2006, 2008) and following the methodology described in Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2017), we also determined accurate masses for the primary stars of these systems, the Ba stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In this third article, we focus on the faint WD companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used the orvara software (Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2021c) to combine all the radial- velocity data available, the astrometric measurements from the Hipparcos mission (Perryman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1997), the Gaia positions and proper motions (Lindegren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2021), and the information in the Hipparcos-Gaia Catalogue of Accelerations (HGCA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Brandt 2018, 2021) to determine the astrometric orbital parameters of as many Ba star systems as possible (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2 for the descrip- tion of the sample), and then derive the mass of the secondary stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' All these data sets are described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' An important improvement with respect to what has been attempted before for these objects is that we use a joint astrometric-spectroscopic model (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 4) to find new best-fitting orbital parameters in- stead of relying only on RV data or imposing the spectroscopic solution on the astrometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Our results are presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 5 and their implications are discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We also discuss the feasibility of the direct detection of the WD compan- ion for a subset of the longest-period systems in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Target selection For our methodology (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 4) to be applicable, a target must fulfil three requirements: (i) it must be part of the HGCA, (ii) we must have a good initial estimate of the mass of the primary star in the system, and (iii) the Hipparcos solution cannot not be more complex than the 5-parameter solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' As a starting point, we selected all the Ba stars from the samples studied by Joris- sen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019), Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b) and North et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2020 that have Hipparcos identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We excluded confirmed triple systems, stars whose Ba star nature was not certain or is un- der current investigation (Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' under review), and a few systems that had an acceleration solution or an orbital solution in the Hipparcos data reduction (solution types, Sn, equal to 7, 9 and 75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We ended up with 60 systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Table 1 presents our target list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In addition to the most com- monly used identifier, we include the Hipparcos identifier of each system and the Ba star type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We distinguish between pre- RGB, which are all the stars classified as dwarfs or subgiants by Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b) and North et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2020), and mBag or sBag which are stars classified as mild or strong Ba giants by Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019) based on their [La/Fe] and [Ce/Fe] values (as measured by Smith 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Allen & Barbuy 2006a,b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Pereira et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Karinkuzhi & Goswami 2014, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Luck 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' de Castro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Merle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Van der Swaelmen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Karinkuzhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019) and on the Ba index introduced by Warner (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The table also lists the Ba star masses (MBa) that we used as a prior in our MCMC model (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 4) and the metallicity of the system, both values col- lected from Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019), Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b) or North et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2020) unless explicitly specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' For this work, we recom- puted the primary masses for the ten systems that were part of the non-single-star (NSS) Gaia DR3 catalogues (Gaia Collabo- ration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We followed the exact same procedure fol- lowed and described in the mentioned papers and used the same STAREVOL grids of models (Siess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Siess & Arnould 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2017), but we used the NSS Gaia DR3 par- allaxes to recalculate their luminosities and masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Finally, the last column of Table 1 includes the sources where we found the archival RV data used in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Radial velocity and astrometric data 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' CORAVEL, HERMES and other radial-velocity data The most important radial-velocity monitoring programs of Ba stars were carried out with the two CORAVEL spectrome- ters and with the HERMES high-resolution spectrograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The CORAVEL spectrometers (Baranne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1979) were installed on the 1-m Swiss telescope at the Haute-Provence Observatory and on the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='54-m Danish telescope at ESO - La Silla, while HERMES (Raskin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Raskin & Van Winckel 2014) is mounted on the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2-m Flemish Mercator telescope at the Obser- vatory El Roque de Los Muchachos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The main results of these radial-velocity programs were pub- lished by Jorissen & Mayor (1988);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (1998);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Udry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (1998a,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' North et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2000);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Gorlova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Joris- sen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b) among others, and the strength of combining the two data sets, particularly for the longest-period systems, was discussed in the last two mentioned papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019) and Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b) also de- scribed the data reduction process for the two instruments and the existence of a non-zero radial-velocity offset between the data sets due to the use of a different system of standard stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Article number, page 2 of 49 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Table 1: List of Ba star systems to which our methodology was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Column 1 lists the most commonly used identifiers, while column 2 lists the Hipparcos identifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Column 3 lists the Ba star type, which can be preRGB for stars classified as dwarfs or subgiants, or mBag or sBag for stars classified as mild or strong Ba giants, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Column 4 lists the primary star masses and column 5, the metallicity of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' These values were derived or collected by Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b) or Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019) for preRGB and giant systems, respectively, unless otherwise indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Finally, the last column gives the sources of the archival RV data we used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' BD/HD HIP type MBa [M⊙] [Fe/H] RV ref∗ HD HIP type MBa [M⊙] [Fe/H] RV ref∗ 10o4311 80356 preRGB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 ± 0.' metadata={'source': 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± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06(0) −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 N20 ∗ RV reference abbreviations: E19: Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b), U98a: Udry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (1998a), M04: Moultaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2004), M90: McClure & Woodsworth (1990), J19: Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019), G96: Griffin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (1996), U98b: Udry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (1998b), J98: Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (1998), J95: Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (1995), G91: Griffin (1991), N20: North et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2020), G06: Griffin (2006) Mass & metallicity references: (0) This work;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (1) Bensby & Lind (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2) Karinkuzhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (3) North et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2020) This zero-point offset depends on the stellar velocity and on the target’s colour B-V, and there is no real consensus about how to treat it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019) derived it after fitting each orbit by minimizing the orbital residuals, while Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b) determined a relation between the offset and B-V by comparing old and reprocessed CORAVEL data and calculated a fixed off- set for each studied Ba star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' For this work, we combined the two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Where the RV data of a specific instrument spanned over a full orbit or more, we treated the offset as an additional free parameter that was optimized during the orbital fitting pro- cess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' However, for systems with very few HERMES points or for some very long orbits, the offsets from Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019) or Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b) were adopted and fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This will be clearly indicated in the captions of each RV fit shown in Ap- pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Future monitoring with HERMES would remove the need for this assumption, allowing us to fit the offset term di- rectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' To complement the main CORAVEL and HERMES data, we collected additional radial-velocity measurements from other works and instruments, and the sources are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' An optimizable RV offset, such as the one described above between CORAVEL and HERMES, was considered for each data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Hipparcos astrometric data The Hipparcos satellite ESA (1997), launched in 1989, was the first space mission with precision astrometry as its main goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Between 1989 and 1993, Hipparcos measured the location and motion on the sky of more than 100,000 stars many times, to fig- ure out their astrometric path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' For each target in Table 1, we used the positions and the proper motions from the Hipparcos Cata- Article number, page 3 of 49 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main logue (Perryman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Additionally, we also queried the individual astrometric measurements from the re-reduction of the Hipparcos intermediate astrometric data (IAD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' van Leeuwen 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The coordinates are expressed in the International Celes- tial Reference Frame (ICRF) at the 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Since the code we are using is not yet prepared to deal with Hipparcos solutions more complex than the 5-parameter solu- tions, we excluded a few targets with acceleration or orbital so- lutions from the initial sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Some of our remaining targets have a stochastic Hipparcos solution (Sn = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' These represent cases where the residuals are significantly larger than expected, but since the proper motions and the IAD were obtained using a 5-parameter solution, we included them and gave them no spe- cial treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Gaia astrometric data The Gaia mission (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2016, 2018, 2021) was launched in 2013 as a successor of Hipparcos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' For now, none of the Gaia Data Releases (DR) published individual astrometric measurements, so we queried the positions and proper motions published for our targets in the Early DR3 catalogue (Lindegren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In contrast with the Hipparcos data, these are ex- pressed in the ICRF at the 2016 epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The Gaia EDR3 paral- laxes were also queried and used as prior in the fit (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Finally, in order to use an equivalent to epoch astrometry, we also used the Gaia Observation Forecast Tool (GOST1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The GOST provides the predicted observations and scan angles for any Gaia source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We note that not all the planned observations will be used in the final astrometric solution, since some pre- dicted scans might correspond to satellite dead times or might be unusable or rejected as outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' For example, up to 20% of the observations predicted by GOST were excluded from the analy- sis published in Gaia DR2 (Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Ten of the 60 targets presented in this study had a non- single-star (NSS) solution in Gaia DR3 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' These targets are: HD 50264, HD 207585, HD 221531, HD 34654, HD 49841, HD 199939, HD 224621 and HD 87080, which had a non-single-star solution compatible with a com- bined astrometric and single lined spectroscopic model, and HD 44896 and HD 123585, which had a solution compatible with an astrometric binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' For these targets, we used the Gaia DR3 NSS parallax as priors, instead of the EDR3 value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Even though the Gaia DR3 NSS catalogue provided orbital inclina- tions for these 10 systems, we decided not to include an incli- nation prior in our calculations to first, treat all systems equally, and second, compare our independently determined inclinations with the new Gaia ones and validate our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The Hipparcos-Gaia Catalogue of Accelerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' As an additional astrometric constraint, we used the difference in Hipparcos and Gaia proper motions via the Hipparcos-Gaia Catalogue of Accelerations (HGCA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Brandt 2018, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This catalogue puts the Hipparcos, Gaia, and Hipparcos-Gaia (H-G) proper motions into the same reference frame to make them suit- able for orbital fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The Hipparcos-Gaia proper motion is de- rived from the right ascension and declination measurements from the two missions and is by far the most precise due to the long time elapsed between them (proper motion uncertain- ties scale inversely with the time baseline).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This acceleration in the inertial frame can be used to improve the dynamical parame- 1 https://gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='esac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='int/gost/ ters of the companion and to measure its mass because it breaks the mass-inclination degeneracy that RV data suffers from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used the EDR3 version of the HGCA (Brandt 2021) for all our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The EDR3 version of the HGCA also provides a χ2 value be- tween the two most precise proper motion measurements (nor- mally EDR3 and H-G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This value is meant to find accelerating candidates for follow-up and if it is higher than ∼11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 (Brandt 2021), the measured acceleration is considered significant and statistically different, by 3σ, from constant proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In our case, since all our targets are known binaries, we do not need this χ2 value to detect accelerators, but it can give us a hint about which systems are truly benefiting from the HGCA measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The queried HGCA χ2 values are included in the last column of our result table (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Orbital analysis with orvara Orvara, developed by Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2021c), is designed to si- multaneously fit a single Keplerian model to any combination of radial velocity data and relative and absolute astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The combination of these different data sets, using Orvara or not, has recently proven to be very powerful to improve the accuracy of orbits and to measure precise companion masses, even for very long period systems where the observations only cover part of the orbit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Kervella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2021c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Venner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Franson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Leclerc et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Orvara integrates the Hipparcos and Gaia intermediate as- trometry package (htof;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Brandt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2021b) to fit the Hipparcos epoch astrometry and the times and scan angles of individual Gaia epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The code uses a parallel-tempering Markov chain Monte Carlo method (ptmcmc, Foreman-Mackey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2013) and first fits the RV data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Orvara allows RV points from each instru- ment to have a different RV zero point, which we need at least for the CORAVEL-HERMES combination as discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Then the absolute astrometry is included and fit for the five astrometric parameters (positions, α and δ, proper motions, µα and µδ, and parallax, ϖ) using htof at each MCMC step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' On top of the five astrometric parameters, we fitted the six Keple- rian orbital elements (semimajor axis, a, eccentricity, e, time of periastron passage, T0, argument of periastron, ω, orbital incli- nation, i, and longitude of the ascending node, Ω), the masses of the two components (MBa and MWD), and a radial-velocity jitter per instrument to be added to the uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Note that while the difference between the Hipparcos and Gaia reference frames is taken into account in the HGCA, this is not the case for the IAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' However, the rotation difference in the proper motions is w = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='120, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='173, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='090) mas/yr (Fabricius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' These values are very small compared to the amplitudes of the proper motion curves that we are measuring (of the order from a few to a couple of tens mas/yr, see Appendix A), and smaller than the residuals to these fits in most cases, so we did not take them into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' For this work, we assumed uninformative priors for the or- bital elements and for the WD mass, but we adopted Gaussian priors for the primary mass and for the parallax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' For each target, we used the MBa value given in Table 1 but using three times the error bar as sigma to be conservative and take into account systematic errors not accounted for in the statistical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Concerning the parallax, the Gaia EDR3 value was used as prior for most targets, and the Gaia DR3 NSS parallax was used for the 10 targets with a NSS solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used 15 temperatures and for each temperature we use 100 walkers with 100,000 steps per Article number, page 4 of 49 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In a few cases, we needed to run twice as long or repeat the calculations using an educated starting position based on our knowledge about the systems from the RV-only fits published by Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019) or Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b), however, in most cases, the MCMC chains converged quite quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We discarded the first 300 recorded steps (the first 15000 overall, as we saved every 50) as the burn-in phase to produce the results presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' For more details about the computational implementation in orvara and htof and for case studies showing the performance of the code, we refer to the mentioned publications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Results Table 2 lists the obtained astrometric-spectroscopic orbital pa- rameters, the best-fitting WD masses, the χ2 of the best fit, and the HGCA χ2 values discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' To make the ta- ble easier to read, we assume that the error bars we obtained from the MCMC fit are symmetric and listed only the largest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This means that in some cases, the table lists an overes- timated uncertainty in one of the two directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The χ2 values are an overall absolute astrometric χ2, computed adding the χ2 for the Hipparcos proper motions (χ2 H), the χ2 for the long-term Hipparcos-Gaia proper motions (χ2 HG), and the χ2 for the Gaia proper motions (χ2 G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' orvara uses RV jitter terms such that the reduced χ2 of the RV fit is 1, so we did not take it into account to evaluate the goodness of the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The table is ordered based on the orbital period, with the systems with the longest periods first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This way, we can notice that all the systems with periods longer than ∼ 3 years have sig- nificant astrometric accelerations according to their HGCA χ2 values, while most of the systems below that threshold do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Finding such a clear threshold in a sample of confirmed binaries is an indication of the type of systems that the HGCA can help identify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In addition to the table and in order to illustrate and dis- cuss how the results that we get from orvara look like, we include the results for the main-sequence Ba star HD 2454 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' HD 2454 was first identified as a Ba dwarf by Tomkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (1989), and North et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2000) confirmed its binarity even though they did not have enough data to estimate its orbital pe- riod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' More recently, Gray et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2011) found direct evidence of the presence of a WD companion in the system thanks to the Galaxy Evolution Explorer (GALEX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Martin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2005) UV ob- servations and, since 2011, HD 2454 has been part of the long- period binary monitoring program carried out with the HER- MES spectrograph (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In spite of having almost three decades of RV data between the CORAVEL and HERMES mea- surements, Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b) were not able to constrain the orbit either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' However, combining all these RV data points with the Hipparcos and Gaia information, we can finally estimate the orbital elements of HD 2454 as well as the mass of its WD com- panion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1 shows, on the top left panel, the astrometric orbit of HD 2454, including the predicted position of the companion on the scheduled date of Gaia DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The best-fitting orbit is plot- ted as a black thick line, while 40 other well-fitting orbits are colour-coded as a function of the companion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' On the top right panel, we show the RV curve of HD 2454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' For this target we had CORAVEL (orange circles), SOPHIE (pink diamonds), and HERMES (green triangles) RV data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The plot shows that leaving the RV offsets between instruments completely free pro- duces families of solutions with similar orbits and masses but different RV offsets (displaced vertically in the RV plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This is especially noticeable in cases like this one, where no data sets covers even half an orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We want to note that even though we left the RV offsets free in most cases, we always made sure that the best-fitting solution required reasonable values and, espe- cially in the CORAVEL-HERMES case, that these values were close to the values obtained by Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019) and Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The two bottom panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1 show the fit to the proper motions in the right ascension (left) and declination (right) direc- tions, as measured by Hipparcos (squared data point) and Gaia (circular data point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' All the data sets included in the figures were fitted at the same time, and the plotted models are the same in all plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2 shows the one and two-dimensional projec- tions of the posterior probability distributions of the masses of the two components in the system and a few orbital parameters (semi-major axis, eccentricity, and inclination) from the joint RV and astrometric MCMC computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This corner plot shows that the two masses are correlated, and that the semimajor axis is also correlated with the total mass of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' These corre- lations are even stronger for other targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We have included in Appendix A figures similar to Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1 and 2 for all the targets in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Additionally, an individual case of study of a Ba dwarf using the same method was presented in Escorza & De Rosa (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Spectroscopic orbital parameters Even though the main goal of this work was deriving the masses of the WD companions of all these Ba stars, an important addi- tional result of this new method are the new orbits of HD 2454 and BD-11o3853, which could not be constrained before, as well as the improved orbits of a few other long-period systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' When comparing the orbital periods obtained using orvara to those presented in Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019), Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b) and North et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2020), which were obtained by fitting only the RV data, we get a very tight relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The purely spectroscopic pa- rameters and the new parameters are consistent with each other within error bars in almost all cases, and we discuss the excep- tions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' HD 218356 Our first orbital fit for this system converged to a period of more than 40 years, while the period published by Griffin (2006) and Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019) for HD 218356 was 111 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' No third ob- ject has been detected in this system in the past, but the mild s-process enhancement in the visible star has been flagged as surprising given the close orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We performed a three-body fit, setting strong constraints on the inner orbit using the published spectroscopic parameters, and we succeeded to recover the or- bital parameters of two companions, confirming that HD 218356 is actually a triple system with a third companion in a much longer orbit than the published period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The orbital parameters of the system are included in Table 3 and the combined RV fit can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In order to test the significance of this detection, we compared the Akaike Information Criterion (AIC) of the two- and three-component models using the radvel pack- age (Fulton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We found a ∆AIC of 439 favouring the three-component model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Given the masses of the two compan- ions, we expect the WD that polluted the Ba star to be in the outer orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This would also explain the mild s-process enhance- ment reported for HD 218356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We included the corner plots with Article number, page 5 of 49 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Table 2: Orbital elements and WD masses derived following the method described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 4 and listed in order of decreasing periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The columns list, in order, the most commonly used identifier, the orbital period P, the eccentricity e, the time of periastron passage, the absolute semimajor axis of the orbit a, the argument of periastron of the visible star ω∗, the longitudes of the ascending node Ω, the orbital inclination i, and the WD companion mass MWD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' To keep the table cleaner, we assumed symmetric error bars, and included only the largest one of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The last two columns include the χ2 of the best fitting model and the HGCA χ2 value discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Star ID P [days] e T0 [HJD] a [AU] ω∗ [◦] Ω [◦] i [◦] MWD Fit χ2 χ2 HGCA HD 2454 29220 ± 7670 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 2458626 ± 173 22 ± 4 313 ± 19 11 ± 165 34 ± 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='09 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='02 53537 HD 119185 25385 ± 5114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 2477092 ± 5626 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 105 ± 7 136 ± 6 98 ± 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='82 869 BD-11o3853 23376 ± 9862 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 2472263 ± 10323 19 ± 6 199 ± 19 133 ± 10 102 ± 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='46 2534 HD 104979 18518 ± 1205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 2460663 ± 500 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 180 ± 17 34 ± 2 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='087 5851 HD 218356∗ 15194 ± 2630 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13 2469014 ± 2835 22 ± 4 73 ± 24 153 ± 17 157 ± 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='57 388 HD 51959 11195 ± 475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 2459598 ± 329 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 31 ± 11 81 ± 3 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 11044 HD 123949 8544 ± 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9167 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0007 2457772 ± 1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 60 ± 30 122 ± 72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='52 418 HD 182274 8393 ± 51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='039 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='013 2459883 ± 867 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 67 ± 278 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85 65451 HD 18182 8258 ± 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35 2461364 ± 3518 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 194 ± 85 148 ± 46 33 ± 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='46 179 HD 53199 8233 ± 175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='255 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='010 2456826 ± 34 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 66 ± 2 16 ± 3 103 ± 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='51 1873 HD 40430 6147 ± 278 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='02 2457340 ± 54 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 74 ± 5 177 ± 2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='68 4774 HD 95241 5344 ± 55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='807 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='004 2455739.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 108 ± 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='61 7633 HD 139195 5296 ± 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='02 2460005 ± 78 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 29 ± 3 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='27 421 BD-10o4311 4872 ± 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='047 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='006 2456076 ± 105 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 159 ± 8 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 73 ± 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='39 20039 HD 183915 4344 ± 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 2457609 ± 62 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 118 ± 7 77 ± 3 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 5410 HD 180622 4045 ± 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 2458549 ± 3149 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 293 ± 42 166 ± 10 100 ± 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='03 1948 HD 216219 3948 ± 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='085 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='050 2456527 ± 275 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 59 ± 26 52 ± 125 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='73 1059 HD 107541 3583 ± 47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='095 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='035 2458478 ± 3180 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 220 ± 16 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 128 ± 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='092 6173 BD-14o2678 3481 ± 205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 2455846 ± 328 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 283 ± 15 144 ± 11 93 ± 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='28 399 HD 59852 3477 ± 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 2457280 ± 351 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 95 ± 40 139 ± 129 26 ± 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='62 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='52 1772 HD 201824 2922 ± 23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 2456263 ± 102 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 62 ± 9 143 ± 78 59 ± 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='28 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 765 HD 178717 2912 ± 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='03 2455886 ± 57 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 265 ± 5 20 ± 5 35 ± 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13 1357 HD 50082 2883 ± 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='02 2457496 ± 37 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 207 ± 6 22 ± 7 63 ± 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 291 HD 98991 2849 ± 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='323 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='004 2455898 ± 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 5779 HD 205011 2846 ± 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='02 2455297 ± 36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 34 ± 5 56 ± 3 74 ± 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='61 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='36 14392 HD 204075 2367 ± 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 2455474 ± 109 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 255 ± 15 9 ± 165 133 ± 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='62 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 HD 20394 2248 ± 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 2456928 ± 105 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 145 ± 18 92 ± 6 31 ± 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='57 489 ∗For the triple system HD 218356, we only include the outer orbit that hosts the WD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' See Table 3 for the remaining parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Article number, page 6 of 49 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Table 2 continues Star ID P [days] e T0 [HJD] a [AU] ω∗ [◦] Ω [◦] i [◦] MWD Fit χ2 χ2 HGCA HD 16458 2017 ± 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='098 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='025 2456430 ± 112 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 119 ± 14 109 ± 10 61 ± 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='53 3888 HD 5424 1906 ± 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 2455702 ± 145 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 102 ± 14 40 ± 74 30 ± 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='87 445 HD 49641 1793 ± 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 2456298 ± 352 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 207 ± 63 138 ± 126 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 1384 HD 91208 1770 ± 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='178 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='018 2456240 ± 55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 83 ± 10 167 ± 3 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='19 486 HD 200063 1743 ± 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 2456510 ± 120 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 228 ± 27 152 ± 17 115 ± 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='96 4910 HD 201657 1702 ± 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 2456291 ± 280 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 255 ± 61 82 ± 4 153 ± 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='59 1282 HD 43389 1688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='083 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='017 2455663 ± 57 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 190 ± 13 168 ± 150 111 ± 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='03 279 HD 27271 1681.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='224 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='007 2455519 ± 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 210 ± 4 6 ± 2 103 ± 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='09 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='93 1441 HD 95193 1652 ± 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='02 2456007 ± 52 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 285 ± 10 88 ± 37 81 ± 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='33 268 HD 210946 1521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='109 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='011 2455737 ± 35 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 193 ± 8 15 ± 9 114 ± 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 231 HD 127392 1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='088 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='017 2456222 ± 91 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 159 ± 20 4 ± 175 119 ± 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='26 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='31 1531 HD 143899 1461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 2456481 ± 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 279 ± 8 88 ± 3 125 ± 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 640 HD 88562 1451.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='203 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='013 2455931 ± 25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 352 ± 8 61 ± 97 87 ± 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='84 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 HD 221531 1402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='165 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='005 2455563 ± 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 189 ± 3 136 ± 3 59 ± 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='49 794 HD 202400 1391 ± 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='09 2455555 ± 96 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 37 ± 295 86 ± 86 61 ± 67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='48 564 HD 107574 1384.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='083 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='005 2456521 ± 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 216 ± 4 22 ± 2 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 3766 HD 58121 1217 ± 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='135 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='019 2455339 ± 50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 89 ± 10 58 ± 28 121 ± 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 149 HD 31487 1063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='037 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='018 2455808 ± 98 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 237 ± 33 48 ± 2 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='28 1064 HD 211594 1018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='058 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='013 2455675 ± 36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 76 ± 13 65 ± 17 123 ± 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='02 HD 34654 976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1114 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0016 2455212 ± 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 91 ± 4 74 ± 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='058 183 HD 92626 921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='014 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='014 2455685 ± 237 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 116 ± 199 92 ± 62 85 ± 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='61 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='47 HD 50264 910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='077 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='018 2455933 ± 43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 238 ± 17 63 ± 17 103 ± 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='53 246 HD 49841 897.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='162 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='015 2455518 ± 21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 350 ± 6 110 ± 51 109 ± 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='51 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 HD 58368 673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='02 2455715 ± 28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='17 18 ± 6 99 ± 68 78 ± 27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='98 HD 207585 671.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='03 2455613 ± 174 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 109 ± 207 103 ± 17 93 ± 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 HD 44896 629.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='019 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='013 2455388 ± 76 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 228 ± 35 101 ± 35 78 ± 35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='99 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='86 HD 199939 585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='39 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='281 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='012 2455207 ± 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 48 ± 3 110 ± 84 82 ± 21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='61 HD 123585 460.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 2455534 ± 152 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 191 ± 73 27 ± 135 90 ± 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='53 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 HD 24035 378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='014 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='016 2455307 ± 184 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 204 ± 110 52 ± 106 71 ± 31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='74 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='69 HD 224621 308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='020 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='012 2455264 ± 32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 309 ± 57 64 ± 27 31 ± 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 HD 87080 274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='162 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='016 2455230 ± 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 128 ± 5 149 ± 112 60 ± 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='58 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 HD 121447 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='012 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='010 2455243 ± 51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 267 ± 104 90 ± 50 59 ± 76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='71 HD 77247 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5371 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='108 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='005 2455236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='03 40 ± 3 60 ± 35 98 ± 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='31 Article number, page 7 of 49 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1: Orvara results for the main-sequence Ba star HD 2454.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Top: astrometric and spectroscopic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The RV plot includes radial-velocity measurements from CORAVEL (orange circles), SOPHIE (pink diamonds), and HERMES (green triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Bot- tom: Hipparcos and Gaia proper motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In all plots, the best-fitting orbit is plotted as a black thick line, while 40 other well-fitting orbits are included and colour-coded as a function of the companion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' the parameters of both orbits in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Only the outer orbit information is listed together with the other WD orbits in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' HD 201657 Our orbit fit for HD 201657 converged to twice the published orbital period and to a much more eccentric orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The astromet- ric data favours the longer orbit, and the RV data is not very constraining since we have only 15 CORAVEL points and one HERMES point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' However, given the eccentricity-period diagram of Ba stars, the orbit published by Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019), the least eccentric of the two, is the most likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We attempted to recover this orbit in order to check the quality of such a fit and calcu- late the WD companion mass by including an orbital eccentricity prior of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We recovered Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019)’s orbital solution, although with a slightly higher χ2 for the astrometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Since we considered this solution more likely for a Ba star, we listed this orbit in Table 2, but we show both fits and corner plots in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' More HERMES data would be essential to solve this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Astrometric orbital parameters Finally, in addition to the new and improved orbital parameters, this method provided us with orbital inclinations for all these Ba star systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 4 shows the distribution of the obtained cos(i) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This distribution should be flat if we could assume our sample of binaries is randomly distributed on the sky, and even though we only have 60 systems, the distribution is compatible with a uniform one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We performed a Kolmogorov-Smirnov (KS) test (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Press et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1986), and we obtained p-values higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 when comparing our cos(i) distribution with uniform distri- butions of the same sample size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The new orbital parameters are also compatible with the as- trometric parameters published in Gaia DR3 for the ten targets available in their catalogue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Concerning the periods, all Gaia DR3 values are consistent with our values within 2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The largest difference is found for HD 221531, for which Gaia DR3 pub- lished a period of 1668 ± 135 days, about 260 days longer than our period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The Gaia DR3 time span is about 1000 days (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2022), while our data covers a few decades in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Hence, we think that our method is more reliable to obtain the periods of long-period binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The eccentricities Article number, page 8 of 49 3000 CORAVEL HERMES SOPHIE 2000 Mcomp(Mo) RV (m/s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 1000 大 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 2027 O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35 + f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1980 1990 2000 2010 2020 20 Aα (arcsec) Epoch (yr) HIPPARCOS HIPPARCOS 65 180 GAIA GAIA 60 185 55 (mas/yr) /yr) 190 (mas/ 50 n 45 195 40 200 35 205 30 L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1 O-C 0-0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (year) Epoch (year)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='503+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='086 20 30 40 50 60 a (AU) a (AU) = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='588+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 32 40 48 56 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 Msec (M ) 20 30 40 50 60 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72 e 32 40 48 56 i ( ) i ( ) = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2: Corner plot of some derived parameters for HD 2454 including mass of the two stars, the semimajor axis, the eccentricity, and orbital inclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' are compatible as well, without significant exceptions, and fi- nally, we used the Thiele-Innes elements published in the Gaia DR3 catalogue and followed Halbwachs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2022) to compute the orbital inclinations of these systems from the Gaia DR3 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The Gaia DR3 inclinations are also compatible with the inclina- tions we obtained with our full RV+astrometric model within 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 times our σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' As discussed above, the HGCA is not very con- straining for systems with periods below about 3 years, so while we think our method is better to determine the orbital periods of Ba stars, the Gaia DR3 inclinations are probably of better qual- ity than ours for the shorter-period systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' When the epoch as- trometry of the Gaia mission is published, we will be able to combine these data with all our other data sets and improve our results for the shortest period systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' White Dwarf masses Table 2 lists the masses we obtained for the companions to all the Ba stars in our sample, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 5 shows the distribution of these masses as a purple dashed histogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Also in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 5 we compare this new distribution to the distribution obtained by Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019) and Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019a) for the same stars, which is drawn in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The insert in the figure shows the cumulative frequency of the two distributions, including an envelope with the 1 − σ uncertainty for our distribution, which also envelopes the old distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We obtained a p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='010 on a KS test, Table 3: Orbital parameter of the triple system HD 218356 Parameter Inner orbit Outer orbit Period, P [days] 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='03 15194+2600 −1600 Eccentricity, e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='072+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='048 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='39+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' of periastron, T0 [HJD] 2455289+15 −85 2469014+2800 −2800 Semimajor axis, a [AU] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 Arg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' of periastron, ω [◦] 55+270 −37 73+21 −24 Ascending node, Ω [◦] 90+60 −62 153+14 −17 Inclination [◦] 90+42 −41 157+4 −5 Companion mass [M⊙] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 which is not low enough to reject the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The two distributions are not statistically different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 6, we plot the mass distributions of the companions to strong and mild Ba giants, separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' As mentioned in the intro- duction, this distinction is made based on the abundance ratios [La/Fe] and [Ce/Fe] and on the Ba index introduced by Warner (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We do not include the pre-RGB stars in this comparison, because the distinction between strong and mild enhancement Article number, page 9 of 49 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 3: Best fitting models to the RV data of HD 218356 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 4: Distribution of cos i where i are the orbital inclinations of the Ba star systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Bin-width chosen to roughly follow the Freedman–Diaconis rule (Freedman & Diaconis 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' has not been as clearly established as it has for the giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We note that the WDs occupying the high-mass tail belong to sys- tems with strong Ba giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' However, we performed a KS test, and we obtained a p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45, meaning that we cannot re- ject that the two samples are drawn from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The cumulative distributions plotted in the insert also show that Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 5: Mass distribution of WD companions of Ba stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The purple histogram corresponds to the WD masses obtained for this publication, while the black histogram includes the results published in Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019) and Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019a) as- suming a narrow distribution of Q and MWD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The bin-width was chosen to roughly follow the Freedman–Diaconis rule (Freed- man & Diaconis 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The insert in the figure shows the cu- mulative frequency of the same two samples, including a 1-σ envelope for our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 6: Mass distribution of WD companions of strong (blue) and mild (dashed green) Ba giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The insert in the figure shows the cumulative frequency of the same two samples, including a 1-σ envelope for our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' taking the 1 − σ uncertainty into account, the distributions are not very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' There are a few individual systems that appeared as clear outliers or that even have WDs with unphysical masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' These are briefly discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Article number, page 10 of 49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Probability density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 cos(i)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Jorissen19 & Escorza19 This work 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 Probability density 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 M/Mo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 M/moWD with strong Ba giants 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 WD with mild Ba giants 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 Probability density 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 M/Mo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 M/MoCORAVEL 3000 DAO HERMES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2000 (s/w) 1000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 RV 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1000 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3000 2500 O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0 2500 1980 1990 2000 2010 2020 Epoch (yr) 4000 3000 3000 2000 RV (m/s) 1000 (m/s) 2000 0 RV 1000 1000 0 2000 3000 1000 2500 2500 O-C O-C 0 0 2500 2500 1982 1984 1986 1988 2002 2003 2004 2005 2006 Epoch (yr) Epoch (yr)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The least massive WDs: HD 18182 and HD 95241 There are two systems for which our simulations converged to very low WD masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' These are HD 18182 and HD 95241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The fit we achieved for the former is less than ideal (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16), and even though the mass is small, taking the error bars into account, the value is compatible with an average WD in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The CORAVEL RV data is not very constraining and the HERMES points, being of much higher quality, still fall on the same range of orbital phases, covering in total less than half of the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Additionally, the Hipparcos and Gaia proper mo- tions in the right ascension direction are very similar, not adding strong constraints to the fit either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This WD mass should be taken with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The fit for HD 95241, on the other hand, is significantly bet- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used 97 RV points that cover very well the whole orbit (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='21) and obtained clean and symmetric posterior dis- tributions (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Of course, MBa and MWD are very strongly correlated, so if the MBa prior was incorrect, too small in this case, it would directly affect MWD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The mass of HD 95241 was determined by Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b) by comparing the loca- tion of the star on the HR diagram with STAREVOL (Siess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Siess & Arnould 2008) evolutionary tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The stellar pa- rameters were determined from HERMES high-resolution high- signal-to-noise spectra and are in agreement with other studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Takeda 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Soubiran et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' However, HD 95241 was flagged as a mild Ba dwarf by Edvardsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (1993) hav- ing only a marginal overabundance of s-process elements with respect to iron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Other Ba dwarf candidates of their sample have been proven to be wrongly flagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Most of them are likely sin- gle stars (Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' It is possible that HD 95241 has a low-mass companion that is not a WD, and if it is a WD, its AGB progenitor was not massive enough to reach the thermally pulsing AGB phase and produce s-process elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' HD 95241 is likely not a Ba star and will be removed from further analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The most massive WDs: HD 49641 and HD 31487 On the high-mass end of the distributions, there are two systems with WD masses clearly outlying from the initial mass distri- bution (MWD ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 M⊙ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' These are HD 49641, with MWD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 M⊙ , and HD 31487, with MWD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='22 M⊙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The fit for HD 49641 is not very good, because the available RV data was scarce and old, so one should take this WD mass with caution, but the fits for HD 31487 seems reliable, including a clean result for the orbital projection on the sky (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In order to try to explain this last mass, one could again try to in- voke a wrong MBa prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used the primary mass determined by Karinkuzhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The primary mass listed by Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019) is not in agreement with Karinkuzhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2018)’s within error bars, but we decided to use the latter after study- ing their HR diagram (their Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In any case, Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019)’s mass is higher, and would result in a higher companion mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Karinkuzhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2018)’s value seems reasonable given the location of the star on the HR diagram, and it is a very aver- age value for giant Ba stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Additionally, there is no big discrep- ancy between the parallaxes published in the different Gaia Data Releases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' While a wrong parallax could have led to a wrong lu- minosity, hence mass, determination, we have no good reason to doubt this mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' From the posterior distributions and 1D projec- tions shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='87, one can see that a significantly lower MBa could lower MWD within the Chandrasekhar limit (about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 M⊙ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Chandrasekhar 1939), but that the dynamics of this sys- tem do not favour a secondary mass below ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 M⊙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='01 (arcsec) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='015 [arcsec] 2022 Astrometric Orbits 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mcomp(M ) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 7: Projection on the sky of HD 31487.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Only with the dynamical information that we currently have, it is difficult to confirm that this ’massive companion’ is a single object, and not a close pair formed, for example, by a faint main- sequence star and a WD (see van den Heuvel & Tauris 2020 for an example of such a situation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The strong s-process enhance- ment strongly suggests that there is a WD in the system, but since we cannot be certain of its mass, HD 31487 will be re- moved from further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Discussion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Mass distributions The mass distribution that we obtained for the WD companions of Ba stars is compatible with current estimates for field WD masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The average mass of DA WDs (WDs with only Balmer lines in their spectra) is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 M⊙ , while that of DB WDs (WDs with no H or metals lines in their spectra, only helium lines) is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='68 M⊙ (Kleinman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The weighted average of our mass distribution is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 M⊙ , after removing the two tar- gets mentioned in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' There is a high-mass tail present in the mass distribution of WDs orbiting Ba giant that Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019) and Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019a) already discussed (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In order to evaluate if Q = M3 WD/(MBa + MWD)2 is constant, we computed this value for all our targets and present the average and the standard deviation for each one of the three subsamples separately in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The new distributions are marginally dif- ferent to literature Q distributions (see Table 1 in Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We obtained p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='048 for the strong Ba giants, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='035 for the mild Ba giants and p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='012 for the Ba dwarfs, when we performed KS tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The main difference is that the new distributions are not as narrow as obtained in the past when mod- elling f(m) = Q sin3 i, with f(m) being the spectroscopic mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In order to check if this is caused by the fact that the in- dividual inclination uncertainties play a role now, while an incli- nation distribution was assumed in the past, we calculated new Q distributions removing the 10 and 20% systems with larger uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' All the observed distributions are broader than the literature ones, but not significantly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In Table 4, we have also included the average and standard deviations of the current mass ratios of the three Ba star subsam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The two subsamples of giants show closer values, with the Article number, page 11 of 49 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Table 4: Average and standard deviations of the Q-values, with Q = M3 WD/(MBa + MWD)2, and the mass ratios of strong Ba gi- ants, mild Ba giants and pre-RGB Ba stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Ba type Q-value Mass ratio (q) Strong Ba giant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='054 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='09 Mild Ba giant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='036 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 Ba dwarf 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='091 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 average mass ratio of strong Ba giants being slightly higher than that of mild Ba giants, in agreement with what Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019) reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We perform a KS test and obtained a p-value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='015, which is not low enough to statistically confirm that this difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The average mass ratio of Ba dwarfs is much higher, but the currently known Ba dwarfs are significantly less massive than the giants (Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019b), then accounting for this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A comment on nucleosynthesis predictions It is difficult to make a direct correlation between the WD com- panion mass and the s-process enhancement of the Ba star be- cause many parameters, apart from the WD progenitor mass, strongly affect the final Ba star abundances and the unknowns are still stronger than the observational constraints (see Cseh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2022, for a study of abundances in individual Ba giant systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' For example, the efficiency of the mass transfer and the dilution factor, the ratio between the accreted mass and the mass in the Ba star envelope over which this is mixed in, are major uncertain- ties in our understanding of the formation of Ba stars and will directly affect the final s-process enhancement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Stancliffe 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Of course, the efficiency of the s-process of nucleosynthe- sis in the interiors of AGB stars, which strongly depends on the mass and the metallicity of the star itself, is also a key parameter in order to explain a possible correlation between WD mass and Ba enhancement(e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Busso et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Karakas & Lugaro 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Van der Swaelmen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Additionally, even the number of thermal pulses and third dredge-ups experienced by the AGB star before the mass transfer episode took place will have an effect on the final s-process abundance pattern (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Shetye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2018), as well as mixing and diffusion below the AGB star’s convective envelope will (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Goriely & Siess 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Standard stellar-evolution models do not predict solar- metallicity low-mass AGB stars undergo third dredge-ups (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Cristallo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Karakas & Lugaro 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This limit can go down to 1 M⊙ at lower metallicities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Stancliffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Lugaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Fishlock et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' However, including different additional effects in the models can help, for exam- ple, Weiss & Ferguson (2009) showed that including some over- shooting below the convective pulse, their models could make a 1 M⊙ AGB star undergo third dredge-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Additionally, Shetye et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019, 2021) found several low-mass AGB stars currently undergoing third dredge-ups and their models succeeded to re- produce the s-process overabundance including diffusive mixing at the bottom of the stellar envelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Additionally, according to several studies, the AGB stars that polluted Ba stars need to have masses below 3 M⊙ to be able to reproduce their abundance ra- tios with models (Lugaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2003a, 2012, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Cseh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Karinkuzhi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Figure 8 shows the relation between the metallicity (listed in Table 1) and the obtained WD masses (Table 2) for the preRGB Ba stars (orange circles), the strong Ba giants (blue squares) and the mild Ba giants (green triangles) in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The fig- ure shows an expected correlation between the Ba-type and the metallicity, caused by the fact that the efficiency of the s-process in AGB stars decreases as the metallicity increases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Cseh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' However, there is no obvi- ous correlation between the WD mass and the metallicity, even though the AGB mass directly affects the s-process efficiency as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The least massive WDs are in systems with [Fe/H] < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1, in agreement with the models, and the most massive WDs ac- company Ba giants of [Fe/H] between −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2, with the three most massive WDs being in a strong Ba star systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Among our sample of 58 systems (after having removed (HD 95241 and HD 31487 from the WD sample), we do not find Ba stars with unexpectedly low mass companions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' As discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3, the companion mass for HD 18182 should be taken with caution, but all other Ba star systems have WDs of around or more massive than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 M⊙ , meaning that their progenitors were AGB stars of around or more massive than 1 M⊙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Note that to make such a statement, one needs to rely on initial-final mass relationships (IFMR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used as a reference the relation pub- lished by Marigo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Using the same relation, we can claim that a fraction of the AGB stars that polluted our sam- ple of Ba stars were more massive than the expected 3 M⊙ limit, since we found that several WDs have masses around or higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 M⊙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This is the case even taking into account the kink that Marigo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2020, 2022) find for WDs of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 M⊙ with carbon AGB progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Most IFMRs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Wei- demann 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Kalirai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Andrews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Cummings et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' El-Badry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2018) flatten at around MWD ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 M⊙ , making stars with a wide range of initial masses accumulate at that WD mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' However, their progenitors are expected to have initial masses in the range between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 M⊙ , hence more massive than what the Ba stars abundance ratios seem to indicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The presence of these massive WDs orbiting around both strongly and mildly polluted Ba stars presents important con- straints, as well as an interesting challenge, for evolutionary and nucleosynthesis models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Future studies of these systems follow- ing the line presented by Stancliffe (2021) or Cseh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2022), but using these new WD masses, might be able to tell us new things about AGB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We note that our error bars are signif- icant and that these statements blur if we consider two or three sigma uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This will improve when we have NSS paral- laxes to obtain more accurate Ba star masses and Gaia astromet- ric epochs to improve the RV+astrometry fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Direct imaging ob- servations could also help constrain the longest-period systems better (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Future observational prospects: direct imaging of these white dwarfs with SPHERE The nearby (d ≲ 100 pc) Ba stars that are host to long-period (P ≳ 10 yr) companions are suitable candidates for high-contrast imaging observations to spatially resolve the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' These observations would provide relative astrometric and photometric measurements between the WD and the Ba star host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A single measurement of the instantaneous angular separation between the components would constrain both the total semi-major axis, and thus the total system mass, and the inclination (unless the observation occurred when the companion was crossing the line of nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Photometric measurements of the companion could be used to estimate the bolometric luminosity of the compan- ion which, in conjunction with the mass, can be compared to Article number, page 12 of 49 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 8: Relation between the metallicity (Table 1) and the ob- tained WD masses (Table 2) for the preRGB Ba stars (orange circles), the strong Ba giants (blue squares) and the mild Ba gi- ants (green triangles) in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' WD cooling models to estimate the age of the companion (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Bonavita et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2020 for the discovery and analysis of a WD companion around a K-type star with SPHERE and e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Grat- ton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2021 for the study of a sample of Sirius-like systems, long-period main-sequence + white dwarf binaries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We assessed the feasibility of spatially resolving the compan- ion by comparing the predicted angular separation and flux ratio between the WD companion and the Ba host star to the perfor- mance of the VLT/SPHERE instrument (Beuzit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We filtered the sample to exclude systems with a median apoastron distance within 100 mas, calculated from the MCMC samples described in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' For these systems, the companion would always be within the inner working angle of the instrument, and impossible to resolve with SPHERE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This filter resulted in a sub- sample of eight systems for which the companion will be at a projected separation of ρ > 100 mas at some point in its orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The feasibility of direct detection also depends on the flux ratio between the WD companion and Ba star host.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We esti- mate the H-band flux ratio for each MCMC sample using pure- hydrogen (DA) atmosphere mass-luminosity relations from Hol- berg & Bergeron (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We assigned an age to each MCMC sample at random from a uniform distribution between 106 and 1010 yr to account for the unknown age of the WD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The model grid was linearly interpolated in (log t, M) to extract an absolute H-band magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This was converted into a flux ratio using the parallax from the MCMC sample and the apparent H-band mag- nitude of the Ba star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We assumed the companion has negligible flux in the H-band relative to the Ba star, such that the catalogue H-band magnitude of the system can be entirely ascribed to the Ba star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We converted the orbital elements to the angular separation between the WD and the Ba star host at the epochs 2023, 2024, and 2025 for each MCMC sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The predicted angular sepa- ration and flux ratio for each sample was then compared to the SPHERE contrast curve given in Wahhaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We ac- counted for the degradation in contrast performance for fainter stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2022) by scaling the contrast curve by the square root of the H-band flux ratio between the Ba star host and HR 8799, the star observed by Wahhaj et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2021) to measure the contrast curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The predicted separations in 2023 and H-band contrast for each of the eight systems are shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' There are six systems with a non-negligible probability of detection at this epoch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' the others are too faint to be detected given the expected contrast curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The majority of these systems exhibit a strong correlation between the separation at the 2023 epoch and the mass of the WD companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This can partly be explained by the constraint provided by a direct measurement of the semi-major axis of the system, leading to a much more precise measurement of the total mass of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Summary and conclusions The WD companions of Ba stars contain important information about the formation of these chemically peculiar stars, about the binary interaction processes that these systems underwent in the past, and about the nucleosynthesis processes that took place in- side their AGB progenitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' However, they are cool, dim, and generally not detected by direct methods, so they have not been studied in detail in the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A few absolute masses had been determined before this work by combining the spectroscopic or- bital parameters of these systems with Hipparcos astrometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' However, most published masses for WD companions of Ba stars were computed by making assumptions on the relation between the masses of the two stellar components in these sys- tems or on their orbital inclinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In this work, we used the software package orvara to com- bine radial-velocity data, Hipparcos and Gaia positions and proper motions through the Hipparcos-Gaia Catalogue of Ac- celerations, and astrometric epoch measurements from the Hip- parcos mission, and determine the astrometric orbital parameters of 60 stars flagged as Ba dwarfs or giants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Using this method, we could constrain the orbits of two long-period systems that could not have been constrained before with RV data only, and we im- proved the orbital solution of a few other systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Orbital incli- nations were also determined for the first time for many of these systems, and finally, including a prior on the Ba star masses, we also derived the mass of the secondary stars in these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Finally, we discovered that HD 218356, one of the shortest pe- riod Ba star systems known, is actually a triple system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We de- termined the parameters of both the inner and outer orbits and the masses of the two components, and it is very likely that the WD companion that polluted HD 218356 is in the outer orbit, explaining the mild s-process enhancement of the Ba giant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The WD mass distribution presented in this work includes all systems published by Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019), Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b) and North et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2020) that had a single-star Hipparcos solution and that were not confirmed triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This mass distri- bution is compatible with field WD mass distributions and with those published before for Ba stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The distribution extends to high WD masses, higher than expected by theoretical models of the s-process of nucleosynthesis that have focused on reproduc- ing the abundance ratios measured on Ba star atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This work brings new observational constraints for these models and an interesting challenge to our understanding of the formation of Ba stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' In order to look at Ba stars with new eyes, we plan future direct imaging observations of six of the longest-period systems with SPHERE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' On the one hand, this data will provide us with a measurement of the instantaneous angular separation between the components of the system, partially breaking the total mass semimajor axis correlation and helping us get more accurate Article number, page 13 of 49 preRGB Ba stars strong Ba giants 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 mild Ba giants 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 Table 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 MwD/Mo, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 [Fe/H], Table 1A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main 8 10 12 14 16 BD-11 3853 HD2454 HD95241 HD98991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 ρ at 2023 (asec) 8 10 12 14 16 ∆H (mag) HD104979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 HD139195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 HD182274 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 HD218356 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 9: Predicted angular separation in 2023 and contrast of the eight systems with median apoastron distances of > 100 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Contours indicate 1, 2, and 3σ credible regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The predicted SPHERE contrast is given by the solid red line, and the red shaded region corresponds to the inner working angle of the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Six of the systems are amenable to direct detection in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' On the other hand, we will be able to estimate the bolometric luminosity of the WD, which combined with its mass, can be compared to WD cooling models to estimate the age of these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The authors thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Alain Jorissen and Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Henri Boffin for the enriching discussions and Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Hans Van Winckel for pro- viding us with a few new, unpublished HERMES RV points for our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We also want to thank the referee, Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Carine Babusiaux, for helping us to im- prove this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We are grateful to all observers of the HERMES con- sortium for their time dedicated to the Mercator-HERMES long-term binary- monitoring program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The HERMES spectrograph is supported by the Fund for Scientific Research of Flanders (FWO), Belgium, the Research Council of KU Leuven, Belgium, the Fonds National de la Recherche Scientifique (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='-FNRS), Belgium, the Royal Observatory of Belgium, the Observa- toire de Genève, Switzerland and the Thüringer Landessternwarte Tautenburg, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This publication includes data retrieved from the SOPHIE and the ELODIE archives at Observatoire de Haute-Provence (OHP), available at http: //atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='obs-hp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='fr/sophie and http://atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='obs-hp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='fr/elodie, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' This work makes use of the "Synthetic Colors and Evolutionary Sequences of Hydrogen- and Helium-Atmosphere White Dwarfs" hosted at http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='umontreal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='ca/~bergeron/CoolingModels and of the SIMBAD database, operated at CDS, Strasbourg, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Last but not least, this work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='int/gaia), processed by the Gaia Data Process- ing and Analysis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='int/web/ gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Funding for the DPAC has been provided by na- tional institutions, in particular the institutions participating in the Gaia Multi- lateral Agreement.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1: RV curve and proper motions of HD 2454 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2: RV curve and proper motions of HD 119185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 500 m/s (Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='503+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='086 20 30 40 50 60 a (AU) a (AU) = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='588+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 32 40 48 56 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 Msec (M ) 20 30 40 50 60 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72 e 32 40 48 56 i ( ) i ( ) = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3: Corner plot of HD 2454 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='651+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='060 24 32 40 a (AU) a (AU) = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='611+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='070 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 Mpri (M ) 75 90 105 120 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) 24 32 40 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 e 75 90 105 120 i ( ) i ( ) = 98+10 13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4: Corner plot of HD 119185 Article number, page 17 of 49 3000H CORAVEL HIPPARCOS HIPPARCOS 65 E 至 180 SOPHIE GAIA GAIA HERMES 60 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 2000 185 55 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 (mas/yr) (uk/sew) RV (m/s) 190 1000 50E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0r 195 45 E oF 40E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 200 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 35 205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 30 1E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35 O-C +T 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5E 1990 2000 2010 2020 1990 1995 20002005 2010 2015 19901995 2000 2005 20102015 Epoch (yr) Epoch (year) Epoch (year)4000 CORAVEL ■HIPPARCOS 28 IHERMES GAIA 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 30000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 29 [ 2000 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85 (mas/yr) (mas/yr) 1000F RV (m/s) 30 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='80 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 31 1000E 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 2000E 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 3000 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 HIPPARCOS 33 GAIA 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 1000斤 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 日 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1000 1990 2010 19901995 2000 2005 20102015 1990 2000 2020 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5: RV curve and proper motions of BD-11o3853 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6: RV curve and proper motions of HD 104979 Mpri (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='76+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 25 50 75 100 125 a (AU) a (AU) = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='460+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Mpri (M ) 75 90 105 120 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 Msec (M ) 25 50 75 100 125 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 e 75 90 105 120 i ( ) i ( ) = 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7: Corner plot of BD-11o3853 Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='69+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='94+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 15 18 21 24 a (AU) a (AU) = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='116+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 144 146 148 150 152 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) 15 18 21 24 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 e 144 146 148 150 152 i ( ) i ( ) = 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8: Corner plot of HD 104979 Article number, page 18 of 49 CORAVEL HIPPARCOS HIPPARCOS 142 32 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 HERMES GAIA GAIA 2000 144 34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 o上 (uk/sew) (μk/sew) 146 2000 36 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 /w) 148 4000 0nl 38 150 6000 40 152 8000F 42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 2500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 E 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 O-C O-C AA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)2000 60 E HIPPARCOS DAO HIPPARCOS 212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 F GAIA CORAVEL GAIA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 HERMES 55 1000 215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 (mas/yr) 50 (k/se) (s/w) 0 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0E 45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1000 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0E 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 F 2000 35上 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 882- 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2000 1980 2000 1990 2010 2020 1990 1995 2000 2005 20102015 1990 1995 2000 2005 2010 2015 Epoch (year) Epoch (year) Epoch (yr)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9: RV curve and proper motions of HD 51959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 500 m/s (Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10: RV curve and proper motions of HD 123949 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='509+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='080 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 a (AU) a (AU) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='295+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 Msec (M ) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 e 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 i ( ) i ( ) = 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='82 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11: Corner plot of HD 51959 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='78+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13 9 10 11 12 13 a (AU) a (AU) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='84+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='915 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='918 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='91667+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00066 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 60 80 100 120 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) 9 10 11 12 13 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='915 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='918 e 60 80 100 120 i ( ) i ( ) = 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 72 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12: Corner plot of HD 123949 Article number, page 19 of 49 24 1500F 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 26 1000 4 μα* (mas/yr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 (μk/sew) RV (m/s) 3/ 500 公 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 2 1 (Mo) 30 500 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 1000 CORAVEL HIPPARCOS HIPPARCOS 32 HERMES GAIA GAIA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 2 2 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35 O-C O-C O-C 0 0 0 1000 1990 2000 2010 2020 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)18F HIPPARCOS 7500 E GAIA 16F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 10 5000 E 14E 2500 E (uk/sew) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 000 (mas/yr) 8 (s/u) 12 0 E 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 M 2500 6 1comp(M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=') μα* 会 8F 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 4 6E 7500 4 10000E CORAVEL HIPPARCOS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 2 HERMES GAIA 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 大 O-C O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1990 2000 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 2010 2020 Epoch (yr) Epoch (year) Epoch (year)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13: RV curve and proper motions of HD 182274 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14: RV curve and proper motions of HD 18182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 500 m/s (Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='549+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='060 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='061 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 a (AU) a (AU) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='100 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='039+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Mpri (M ) 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 Msec (M ) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='100 e 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 i ( ) i ( ) = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='49+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='93 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15: Corner plot of HD 182274 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='58 4 8 12 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 30 60 90 120 150 a (AU) a (AU) = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Mpri (M ) 40 80 120 160 i ( ) 4 8 12 Msec (M ) 30 60 90 120 150 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 e 40 80 120 160 i ( ) i ( ) = 33+26 13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16: Corner plot of HD 18182 Article number, page 20 of 49 4000 10 CORAVEL SOPHIE 3000 HERMES 5/ 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 2000E (uk/sew) 0 1000E (uk/sew) (s/u) 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0 1000 85 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 2000F 90 3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 20F HIPPARCOS HIPPARCOS 4000 GAIA GAIA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 1000 O-C O-C O-C 公 0F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 1000 1 E 1990 2000 2010 2020 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)1500 CORAVEL HIPPARCOS 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 HERMES 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 GAIA 1000 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 500 9 μs (mas/yr) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 (s/w), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0E 11 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1500 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2000 1000 2 F 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0 0 1000 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 1990 2000 2010 2020 Epoch (yr) Epoch (year) Epoch (year)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='17: RV curve and proper motions of HD 53199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18: RV curve and proper motions of HD 40430 Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='80 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='642+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='051 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 a (AU) a (AU) = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='66+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='255 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='270 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2549+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Mpri (M ) 70 80 90 100 110 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='80 Msec (M ) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='255 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='270 e 70 80 90 100 110 i ( ) i ( ) = 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='19: Corner plot of HD 53199 Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 a (AU) a (AU) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='44+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='262+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 30 60 90 120 150 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 e 30 60 90 120 150 i ( ) i ( ) = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='36+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20: Corner plot of HD 40430 Article number, page 21 of 49 4000E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0F 3000 E 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2000 E (mas/yr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 (mas/yr) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 RV (m/s) 1000 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 4 oF on 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 mp(Mo) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0E 1000F 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 CORAVEL 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0E ELODIE HIPPARCOS HIPPARCOS 3000 HERMES GAIA GAIA 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 500 O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1990 2000 2010 2020 1990 1995 2000 2005 2010 2015 19901995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)-2 HIPPARCOS 10F 2000 GAIA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1500E 3 11 1000F μα* (mas/yr) (mas/yr) 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 RV (m/s) 4F 500E Mcomp(Mo) 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 5 500F 14F 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 15上 1500 CORAVEL HIPPARCOS 不 HERMES GAIA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 F 1000 1F 0-0 0-0 O-C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 1990 2000 2010 2020 2030 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='21: RV curve and proper motions of HD 95241 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='22: RV curve and proper motions of HD 139195 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='52+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='34+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 a (AU) a (AU) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='805 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='810 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='815 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8071+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0037 1 2 3 4 Mpri (M ) 102 105 108 111 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 Msec (M ) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='800 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='805 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='810 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='815 e 102 105 108 111 i ( ) i ( ) = 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='23: Corner plot of HD 95241 Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='80 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='662+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='050 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 a (AU) a (AU) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='82+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='36 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='318+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='022 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Mpri (M ) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='80 Msec (M ) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='36 e 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 i ( ) i ( ) = 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24: Corner plot of HD 139195 Article number, page 22 of 49 HIPPARCOS 75 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 GAIA 120 2000 80 F 125E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 (u/sew) 85 (mas/yr) oF (s/w) 130 90 Mcomp(Mo) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='42 135 0n 95 AAKA 140 100E 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 CORAVEL OIK 105E 145E HERMES HIPPARCOS SOPHIE GAIA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1000 1E O-C O-C O-C 0 W- - 0 1000 1E 2000 2010 2020 2031 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)8000 Lick HIPPARCOS HIPPARCOS 48 上 GAIA GAIA Camb, 115 DAO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 6000F CORAVEL 46 HERMES 44 (mas/yr) 120 4000上 μs (mas/yr) RV (m/s) 42 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 2000上 Mcomp(Mo) 125 40F n oF 38上 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 130 2000 36 F 34 135 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 5000 O-C O-C 0-0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2 1920 1940 1960 1980 2000 2020 1990 1995 2000 2005 20102015 1990 1995 2000 2005 2010 2015 Epoch (year) Epoch (yr) Epoch (year)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25: RV curve and proper motions of BD-10o4311 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='26: RV curve and proper motions of HD 183915 Mpri (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='52+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 4 5 6 7 a (AU) a (AU) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='07 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0470+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 Mpri (M ) 70 80 90 100 110 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 Msec (M ) 4 5 6 7 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='07 e 70 80 90 100 110 i ( ) i ( ) = 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='27: Corner plot of BD-10o4311 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='87+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='61+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) a (AU) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='408+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='035 1 2 3 4 5 Mpri (M ) 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 e 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 i ( ) i ( ) = 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='29 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='28: Corner plot of HD 183915 Article number, page 23 of 49 CORAVEL 72 6000 CORALIE HERMES 46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 74 4000 76 48 2000 (mas/yr) (ak/sew) RV (m/s) 78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 50 oF 80 52 2000F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 82 4000 84 54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 6000F 86 HIPPARCOS 56 HIPPARCOS GAIA GAIA 88 2500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C 0-0 A 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2500 1995 1990 2000 2010 2020 2030 1990 1995 2000 2005 2010 2015 1990 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)1500 DAO 8E CORAVEL HERMES 1000F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 7 μα* (mas/yr) 500E (mas/yr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 6 RV (m/s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4 E 500 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 3 E 1000F HIPPARCOS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 GAIA 2 2000 2 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 O-C 0-0 O-C 0 0 0 2000 E 1980 1990 2000 2010 2020 2030 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='29: RV curve and proper motions of HD 180622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 500 m/s (Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30: RV curve and proper motions of HD 216219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed HERMES-CORAVEL RV offset of 186 m/s Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019b) Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='81+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='80+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) a (AU) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='079+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='035 1 2 3 4 Mpri (M ) 88 96 104 112 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 e 88 96 104 112 i ( ) i ( ) = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='31: Corner plot of HD 180622 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='631+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='081 6 8 10 12 14 a (AU) a (AU) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='085+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 Mpri (M ) 50 75 100 125 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 Msec (M ) 6 8 10 12 14 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 e 50 75 100 125 i ( ) i ( ) = 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32: Corner plot of HD 216219 Article number, page 24 of 49 6000F CORAVEL HIPPARCOS HERMES 16 GAIA 12E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4000 18 13 2000 (mas/yr) (mas/yr) 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 RV (m/s) 14 oF 22 n 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5 2000 24上 16 4000 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 HIPPARCOS 17 GAIA 28 6000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1000 O-C 0-0 1000 1990 2000 2010 2020 2030 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)HIPPARCOS 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 GAIA GAIA 3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='80 20 A 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0E 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 (μK/sew) *r 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1000 μs (mas/yr) RV (m/s) 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 0 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 10F 1000 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 2000 5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 CORAVEL 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 HERMES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 2000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 F O-C 0-0 O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 2000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1980 1990 2000 2010 2020 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='33: RV curve and proper motions of HD 107541 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='34: RV curve and proper motions of BD-14o2678 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 4 5 6 7 a (AU) a (AU) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='46+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='095+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 50 75 100 125 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) 4 5 6 7 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 e 50 75 100 125 i ( ) i ( ) = 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35: Corner plot of HD 107541 Mpri (M ) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='671+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='094 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) a (AU) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='97+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='253+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='049 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 Mpri (M ) 60 80 100 120 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 Msec (M ) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 e 60 80 100 120 i ( ) i ( ) = 93+16 18 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='36: Corner plot of BD-14o2678 Article number, page 25 of 49 26 CORAVEL HIPPARCOS 4000 GAIA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 14 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2000 16 (μk/sew) (k/sew) 9r RV (m/s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 30 18 on 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2000 32 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 22 4000 34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 O-C O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 F 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 198619881990 1990 1995 2000 1992 1994 1996 2005 2010 2015 1990 1995 2000 2005 20102015 Epoch (yr) Epoch (year) Epoch (year)Φ CORAVEL HIPPARCOS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5上 GAIA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 4000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 W 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2000 (u/sew) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 RV (m/s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 E 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 E 0 on 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2000 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 HIPPARCOS 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 4000 GAIA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0E 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C C 1E 1000 1990 1995 20002005 20102015 1990 1995 20002005 2010 2015 1988 1990 1992 1994 1996 Epoch (yr) Epoch (year) Epoch (year)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='37: RV curve and proper motions of HD 59852 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='38: RV curve and proper motions of HD 201824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 500 m/s (Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='51+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='62+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 4 5 6 7 8 a (AU) a (AU) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='57+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='141+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='083 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Mpri (M ) 20 25 30 35 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) 4 5 6 7 8 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 e 20 25 30 35 i ( ) i ( ) = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='39: Corner plot of HD 59852 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='78+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='28 3 4 5 6 7 a (AU) a (AU) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='49+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='296+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='034 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Mpri (M ) 60 80 100 120 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) 3 4 5 6 7 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 e 60 80 100 120 i ( ) i ( ) = 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1+66 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40: Corner plot of HD 201824 Article number, page 26 of 49 CORAVEL HIPPARCOS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2000 GAIA 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1000 μα* (mas/yr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 (mas/yr) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 RV (m/s) 2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1000 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2000 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 F O-C O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 H 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1990 1990 1995 2000 2005 2010 2015 1988 1992 1994 1996 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)15 24F HIPPARCOS HIPPARCOS GAIA GAIA 6000F Q 14E 22 4000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 13 8 (mas/yr) (u/sew) RV (m/s) 2000 20 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 11 0n 18 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 10 4000 16 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 6000 CORAVEL HERMES 10000 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1 O-C O-C O-C 0F C 0 10000 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2015 1995 1980 1990 2000 2010 2020 1990 2000 2005 2010 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='41: RV curve and proper motions of HD 178717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 500 m/s (Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='42: RV curve and proper motions of HD 50082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 500 m/s (Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='73+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='53+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 3 4 5 6 7 a (AU) a (AU) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='52 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='461+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='028 1 2 3 4 Mpri (M ) 30 60 90 120 150 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) 3 4 5 6 7 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='52 e 30 60 90 120 150 i ( ) i ( ) = 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='43: Corner plot of HD 178717 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 3 4 5 6 7 a (AU) a (AU) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='91 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='189+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='020 1 2 3 4 Mpri (M ) 56 60 64 68 72 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) 3 4 5 6 7 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 e 56 60 64 68 72 i ( ) i ( ) = 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='44: Corner plot of HD 50082 Article number, page 27 of 49 4000 CORAVEL 12 E OIK HIPPARCOS HERMES GAIA 10 F 3000 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 9 2000 (mas/yr) 10E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 (mas/yr) 8 1000 RV (m/s) E 16 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 oE Mcomp(Mo) on μs 6 8E 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 5 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3000 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 2500F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 O-C O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 一 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2500Ei 2005 2015 1990 1980 1990 2000 2010 2020 1990 1995 2000 2010 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)14 F 4000 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 12 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 μα* (mas/yr) 6 μs (mas/yr) 11 RV (m/s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0 HIPPARCOS 10F GAIA 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2000 9E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 8 E 4000 7 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 D CORAVEL HIPPARCOS 6000 HERMES GAIA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 1000 O-C O-0 0 O 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1000 1980 1990 2000 2010 2020 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45: RV curve and proper motions of HD 98991 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='46: RV curve and proper motions of HD 205011 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='43+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='568+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='076 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 a (AU) a (AU) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='96+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='336 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3234+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 123 124 125 126 127 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 Msec (M ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='336 e 123 124 125 126 127 i ( ) i ( ) = 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='88+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='74 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='47: Corner plot of HD 98991 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='79+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='83 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='61+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 a (AU) a (AU) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='69 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='28 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='231+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='024 1 2 3 4 Mpri (M ) 66 72 78 84 90 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='28 e 66 72 78 84 90 i ( ) i ( ) = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='48: Corner plot of HD 205011 Article number, page 28 of 49 290 CORAVEL 6000 HERMES 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 300 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 (mas/yr) 30 (mas/yr) RV (m/s) KAAK 310 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 HIPPARCOS GAIA 40 320 μα* 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 330 50 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 4000 340 60 2500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 O-C 0-0 O-C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 A 一 2500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 E 1990 1995 20002005 2010 2015 1995 2000 2005 2010 2015 2020 1990 1995 20002005 20102015 Epoch (yr) Epoch (year) Epoch (year)6000 每口 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 4000 8 (μ/sew) * 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2000 μs (mas/yr) RV (m/s) 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0 2 Mcomp(Mo) 4 on 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2000 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 4000 0 HIPPARCOS HIPPARCOS DAO 一 HERMES GAIA GAIA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2500 O-C O-C O-C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 2500 1975 1980 1985 1990 1995 2000 2005 2010 1995 1990 1995 2000 2005 2010 2015 1990 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='49: RV curve and proper motions of HD 204075 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50: RV curve and proper motions of HD 20394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 500 m/s (Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Mpri (M ) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='49+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 a (AU) a (AU) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='01+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='257+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='049 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 120 128 136 144 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 Msec (M ) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 e 120 128 136 144 i ( ) i ( ) = 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='51: Corner plot of HD 204075 Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='01+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='491+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 a (AU) a (AU) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='161+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Mpri (M ) 30 60 90 120 150 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 Msec (M ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 e 30 60 90 120 150 i ( ) i ( ) = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='52: Corner plot of HD 20394 Article number, page 29 of 49 DAO HIPPARCOS HIPPARCOS 3000F HERMES 4 F GAIA 28 GAIA 2000E 26F 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1000 (mas/yr) μs (mas/yr) 24 E RV (m/s) 0 22 [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 2 1000 on 20F 2000 4 18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 3000 16 6 2500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C O-C O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 E 2500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 19952000200520102015 19901995200020052010 2015 1980 1985 1990 1995 2000 2005 2010 1990 Epoch (yr) Epoch (year) Epoch (year)CORAVEL 3000 HIPPARCOS HIPPARCOS HERMES GAIA GAIA 3/ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 4 2000 5 2 E 1000 μα* (mas/yr) μs (mas/yr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1 (s/w) o[ RV 1000 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1 2000 8E 3000 2 9F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 4000 3 2500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 F 1F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 O-C O-C O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0 2500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 E 1990 1995 2000 2005 2010 2015 19901995 20002005 1985 1990 1995 2000 2005 2010 2015 2020 2010 2015 Epoch (yr) Epoch (year) Epoch (year)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='53: RV curve and proper motions of HD 16458 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='54: RV curve and proper motions of HD 5424.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 500 m/s (Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='74+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) a (AU) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='098+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Mpri (M ) 60 80 100 120 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 e 60 80 100 120 i ( ) i ( ) = 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7+12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55: Corner plot of HD 16458 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='52+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 a (AU) a (AU) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='69+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='188+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Mpri (M ) 30 60 90 120 150 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 Msec (M ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 e 30 60 90 120 150 i ( ) i ( ) = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56: Corner plot of HD 5424 Article number, page 30 of 49 6000 HIPPARCOS HIPPARCOS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 GAIA 65 GAIA 4000F 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2000 70 μα* (mas/yr) μs (mas/yr) RV (m/s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2000 4000 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 6000 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 CORAVEL 85 8000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1000 O-C 0-0 O-C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1980 1982 1984 1986 1988 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)-8 CORAVEL 32 HIPPARCOS HIPPARCOS 3000 GAIA HERMES GAIA 33 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 34 1000 (mas/yr) 11 (uK/sew) 9r RV (m/s) 12 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 Mcomp(Mo) on 13 1000 36 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2000 37 15 3000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 500 O-C O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1995 1990 1995 2000 2005 2010 2015 2020 1990 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='57: RV curve and proper motions of HD 49641 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='58: RV curve and proper motions of HD 91208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 500 m/s (Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Mpri (M ) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1 2 3 4 Msec (M ) Msec (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='23+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='41 4 8 12 16 a (AU) a (AU) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='91+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='058+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='041 4 8 12 16 Mpri (M ) 40 80 120 160 i ( ) 1 2 3 4 Msec (M ) 4 8 12 16 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 e 40 80 120 160 i ( ) i ( ) = 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='59: Corner plot of HD 49641 Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='29+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='83+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) a (AU) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='19+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='178+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 124 128 132 136 140 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 e 124 128 132 136 140 i ( ) i ( ) = 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60: Corner plot of HD 91208 Article number, page 31 of 49 3 HIPPARCOS HIPPARCOS 3000上 8 E GAIA GAIA 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 2000 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 (mas/yr) 1000F μs (mas/yr) 6 RV (m/s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 12 on 8 mp(M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 1000F 13 9 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 2000F 10 15 3000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 2500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0-0 百 O-C O-C 古 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 2500 1995 2005 2015 2005 1980 1982 1984 1986 1988 1990 2000 2010 1990 1995 2000 2010 2015 Epoch (yr) Epoch (year) Epoch (year)6000 CORAVEL 30 HIPPARCOS HIPPARCOS 10 HO HERMES GAIA GAIA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 32 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 (mas/yr) 2000 μs (mas/yr) RV (m/s) 34 4 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 Mcomp(Mo) 0 on 36 2 E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2000 oF 38 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 2上 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C O-C O-C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1000 2 199019952000200520102015 1990 1995 2000 ¥2005 1990 1995 2000 2005 2010 20152020 20102015 Epoch (yr) Epoch (year) Epoch (year)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='61: RV curve and proper motions of HD 200063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 500 m/s (Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='62: RV curve and proper motions of HD 43389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 500 m/s (Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='44+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='95+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 a (AU) a (AU) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='26+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='073+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='039 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 60 75 90 105 120 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 e 60 75 90 105 120 i ( ) i ( ) = 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 47 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='63: Corner plot of HD 200063 Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='76+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 a (AU) a (AU) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='96+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='083+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='017 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 60 75 90 105 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 e 60 75 90 105 i ( ) i ( ) = 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 30 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='64: Corner plot of HD 43389 Article number, page 32 of 49 5 CORAVEL 6000 HERMES 16 4 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 4000 3 2000 μα* (mas/yr) 14 (mas/yr) RV (m/s) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 13 0 2 2000 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4000 11 0 6000 10 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 HIPPARCOS HIPPARCOS GAIA GAIA 8000 9 1E 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 O-C O-C O-C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0 上 V 500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5E 工 1F 1980 1990 2000 2010 2020 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)HIPPARCOS HIPPARCOS 6000 GAIA GAIA 15 8 4000 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 9 2000 (mas/yr) (mas/yr) RV (m/s) 10 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0 18 Mcomp(Mo) 11 2000[ n μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 12 4000F 20 E 13 6000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 CORAVEL HERMES 14 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 O-C O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1000 1日 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2000 2020 2015 2010 1990 2010 1990 1995 2000 2005 2010 1990 1995 2000 2005 2015 Epoch (yr) Epoch (year) Epoch (year)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65: RV curve and proper motions of HD 27271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='66: RV curve and proper motions of HD 95193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 500 m/s (Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='88+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='700+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='089 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 a (AU) a (AU) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='23+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2236+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0072 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 60 75 90 105 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 Msec (M ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 e 60 75 90 105 i ( ) i ( ) = 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='67: Corner plot of HD 27271 Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='707+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='062 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 a (AU) a (AU) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='133+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='020 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 Mpri (M ) 60 80 100 120 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 Msec (M ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 e 60 80 100 120 i ( ) i ( ) = 81+25 17 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='68: Corner plot of HD 95193 Article number, page 33 of 49 HIPPARCOS HIPPARCOS GAIA GAIA 56 4000 1 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85 58 2000 2 μα* (mas/yr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='80 μs (mas/yr) RV (m/s) 60 0 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 62 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 4 64 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 5 66 6000 CORAVEL VO HERMES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 SOPHIE 68 6 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C O-C O-C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 E 1000 2000 2020 1990 1995 2005 1990 2010 2000 2010 2015 1990 1995 2000 2005 2015 2010 Epoch (yr) Epoch (year) Epoch (year)6000 HIPPARCOS HIPPARCOS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 GAIA GAIA L 4000 0 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85 μα* (mas/yr) 2000 1 μs (mas/yr) RV (m/s) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='80 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 2000 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 4000 5 CORAVEL 14 HERMES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 6000 6 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 O-C O-C 0 0 1000 2 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='69: RV curve and proper motions of HD 210946 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70: RV curve and proper motions of HD 127392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 195 m/s (Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019b) Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='86+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 a (AU) a (AU) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='88+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='150 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1094+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0089 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 60 75 90 105 120 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='150 e 60 75 90 105 120 i ( ) i ( ) = 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='71: Corner plot of HD 210946 Mpri (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='53 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='73+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 a (AU) a (AU) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='150 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='088+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Mpri (M ) 60 75 90 105 120 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='150 e 60 75 90 105 120 i ( ) i ( ) = 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72: Corner plot of HD 127392 Article number, page 34 of 49 6000 HIPPARCOS HIPPARCOS GAIA GAIA 2 4000 2000 2 E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 μα* (mas/yr) μs (mas/yr) RV (m/s) 3 2000 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 5 4000F 6F 8 6000F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 CORAVEL 7E HERMES 8000 SOPHIE 8上 1111 2000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C O-C O-C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5E 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5E, 2015 1990 2000 2010 2020 1990 1995 2000 2005 2010 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)HIPPARCOS HIPPARCOS 0 7500 20 GAIA GAIA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 5000 10F 2500 μα* (mas/yr) 30 (mas/yr) RV (m/s) 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0F Mcomp(Mo) 35 2500上 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 40 5000E 40 7500 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 CORAVEL HERMES 10000| 500 10E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 O-C O-C 0-0 0 500 V 10E 2000 1990 2015 1990 1990 1995 2000 2005 2010 2015 2020 1995 2005 2010 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='73: RV curve and proper motions of HD 143899.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 500 m/s (Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='74: RV curve and proper motions of HD 88562.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 500 m/s (Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='80 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='659+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='055 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) a (AU) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='66+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='181+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='037 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Mpri (M ) 116 120 124 128 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='80 Msec (M ) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 e 116 120 124 128 i ( ) i ( ) = 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75: Corner plot of HD 143899 Mpri (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='99+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='477+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='087 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 a (AU) a (AU) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='203+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 Mpri (M ) 75 90 105 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 Msec (M ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 e 75 90 105 i ( ) i ( ) = 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3+9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='76: Corner plot of HD 88562 Article number, page 35 of 49 6000 HiPPARCOS HIPPARCOs GAIA GAIA 34 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 32 (uk/sew) * 2000 μs (mas/yr) RV (m/s) 2 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 30 Mcomp(Mo) 1 F 0n 28 2000 oF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 26 4000 1 CORAVEL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 不 HERMES 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2 1 O-C O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 O-C 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2 19901995 1990 20002005 2010 20152020 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)10000 HIPPARCOS HIPPARCOS GAIA GAIA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 21 7500 E 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 5000E (mas/yr) 22 μs (mas/yr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 RV (m/s) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 23 0n 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 2500 24 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 5000上 CORAVEL 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 25 HERMES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35 7500 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 O-C 0-0 O-C 0 1990 2000 2010 2020 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='77: RV curve and proper motions of HD 221531 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='78: RV curve and proper motions of HD 202400 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='582+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='086 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 a (AU) a (AU) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='97+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='176 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1653+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 60 80 100 120 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 Msec (M ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='176 e 60 80 100 120 i ( ) i ( ) = 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='79: Corner plot of HD 221531 Mpri (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 a (AU) a (AU) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='87+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='246+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 Mpri (M ) 30 60 90 120 150 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 e 30 60 90 120 150 i ( ) i ( ) = 61+67 15 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='80: Corner plot of HD 202400 Article number, page 36 of 49 HIPPARCOS HO 7500 E 35 GAIA 5 GAIA 5000 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0 10 2500 K (mas/yr) μs (mas/yr) RV (m/s) 25 0E 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 2500 on 20 20 5000 E 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 7500 25 CORAVEL 10000 10 HERMES 30日 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 5 O-C O-C O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0 2500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1990 1990 2015 1990 1995 2000 2005 2010 2015 2020 1995 2000 2005 2010 2015 1995 2000 2005 2010 Epoch (yr) Epoch (year) Epoch (year)15000 HIPPARCOS GAIA 25 55 E 10000 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 μα* (mas/yr) 50 (mas/yr) RV (m/s) 5000 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 45 0 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 40 45 5000 CORAVEL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1 FEROS 35 CORALIE HIPPARCOS SALT-HRS 50 GAIA 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 5000 OTT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C O-C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0 人工 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 5000 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content="5 1990 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 2020 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year):':A." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='81: RV curve and proper motions of HD 107574 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='82: RV curve and proper motions of HD 58121 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='744+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='061 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 a (AU) a (AU) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='99+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='096 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0832+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 Mpri (M ) 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 Msec (M ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='096 e 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 i ( ) i ( ) = 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='31+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='39 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='83: Corner plot of HD 107574 Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='67+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 a (AU) a (AU) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='42+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='135+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='019 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Mpri (M ) 60 80 100 120 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 Msec (M ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 e 60 80 100 120 i ( ) i ( ) = 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='84: Corner plot of HD 58121 Article number, page 37 of 49 135 HIPPARCOS HIPPARCOS GAIA GAIA 20 2000F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85 140 1000 μα* (mas/yr) μs (mas/yr) 15 RV (m/s) 大天 大 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='80 145 0 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 1000 150 2000 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 155 CORAVEL 3000 不 HERMES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 1000 5 O-C O-C O-C 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 1000 5 2010 1990 1990 2000 2020 1990 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)6000 F [DAO] HIPPARCOS HIPPARCOS GAIA GAIA 0 4000F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1 2000 μα* (mas/yr) μs (mas/yr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 RV (m/s) 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 Mcomp(Mo) 口 3 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2500 1 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 O-C O-C O-C 0 中 0F 2500 1 2015 1990 2015 1986 1988 1990 1992 1990 1995 2000 2005 2010 1995 2000 2005 2010 Epoch (yr) Epoch (year) Epoch (year)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85: RV curve and proper motions of HD 31487 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='86: RV curve and proper motions of HD 211594.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 500 m/s (Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='54+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 Msec (M ) Msec (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='59+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 a (AU) a (AU) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='27+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='100 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='037+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='018 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 Mpri (M ) 50 75 100 125 i ( ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 Msec (M ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='100 e 50 75 100 125 i ( ) i ( ) = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='87: Corner plot of HD 31487 Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 a (AU) a (AU) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='058+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='013 1 2 3 4 Mpri (M ) 50 75 100 125 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 Msec (M ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 e 50 75 100 125 i ( ) i ( ) = 123+11 16 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='88: Corner plot of HD 211594 Article number, page 38 of 49 HIPPARCOS HIPPARCOS 6000 GAIA GAIA 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4000 0 μα* (mas/yr) 0 2000 (mas/yr) (s/u) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5 0 5 A 2000 10F (Mo) 4000 10 6000F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 15 DAO 8000 HERMES 15 2000F1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 5A 0-0 0氏 0F 0 0 5 2000E 2000 1980 1990 2010 2020 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)HIPPARCOS 6000 CORAVEL HIPPARCOS 24 16F HERMES GAIA GAIA 4000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 14上 22 μα* (mas/yr) 2000 12 (μ/sew) 9r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 RV (m/s) 20 10 oF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 18 8 2000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 16 6 4000H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 14 4 6000E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C O-C O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1000 1990 1995 20002005 20102015 1990 1995 20002005 1990 2000 2010 2020 20102015 Epoch (yr) Epoch (year) Epoch (year)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='89: RV curve and proper motions of HD 34654 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90: RV curve and proper motions of HD 92626 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='641+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='063 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 a (AU) a (AU) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='36+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1150 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1114+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 Mpri (M ) 64 72 80 88 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 Msec (M ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1150 e 64 72 80 88 i ( ) i ( ) = 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='91: Corner plot of HD 34654 Mpri (M ) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='27 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) a (AU) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='96+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='060 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='014+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='010 2 4 6 8 Mpri (M ) 60 75 90 105 120 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='060 e 60 75 90 105 120 i ( ) i ( ) = 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7+8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='92: Corner plot of HD 92626 Article number, page 39 of 49 HERMES 不 HIPPARCOS 10000E HIPPARCOS A GAIA GAIA 7500E 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 115 A 5000 (uk/sew) 120 (uk/sew) RV (m/s) 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70 2500 125 oF "n 160 130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 2500 4 135 5000E 140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 140E 7500氏 145 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 250F 25 10 O-C T O-C O-C 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 10 250 25 2012 2014 2016 2018 2020 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)CORAVEL HIPPARCOS HIPPARCOS 42 7500 GAIA GAIA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 44H 5000E 4 (mas/yr) 2500 (mas/yr) (s/w) 46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 2 o RV( 48 on 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2500F 0 50 5000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2 7500 52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5E O-C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 500 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2005 2015 1990 1995 2005 2010 2015 1986 1988 1990 1992 1994 2000 1990 1995 2000 2010 Epoch (yr) Epoch (year) Epoch (year)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='93: RV curve and proper motions of HD 50264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='94: RV curve and proper motions of HD 49841 Mpri (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='63+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 a (AU) a (AU) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='077+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 Mpri (M ) 60 80 100 120 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 e 60 80 100 120 i ( ) i ( ) = 104+13 40 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='95: Corner plot of HD 50264 Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='819+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='079 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 a (AU) a (AU) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='82+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='162+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='015 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 Mpri (M ) 30 60 90 120 150 i ( ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 e 30 60 90 120 150 i ( ) i ( ) = 109+15 19 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='96: Corner plot of HD 49841 Article number, page 40 of 49 150 10000 HIPPARCOS HIPPARCOS 70 E GAIA GAIA 140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 60 5000F 130 μα* (mas/yr) 50 (uK/sew) 9r 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 RV (m/s) 120 o 40 110 Mcomp(Mo) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 30 100 5000 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 90 CORAVEL 10 HERMES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 10000 80 SALT-HRS oE 25 F 1000F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 10E O-C O-C O-C 0 0 0 一 1000F 10 1990 1995 2000 2005 2010 2015 2020 1990 1995 200020052010 2015 1990 20002005 1995 2010 2015 Epoch (yr) Epoch (year) Epoch (year)HIPPARCOS HIPPARCOS CORAVEL GAIA 8000F GAIA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 6000 2 4000 (μK/sew) *r 2 (mas/yr) RV (m/s) 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2000上 0 6 oE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 2000E 2 8 4000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 10H 6000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 LM 1986 1988 1990 1992 1994 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='97: RV curve and proper motions of HD 58368 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='98: RV curve and proper motions of HD 207585 Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='57+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='66+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 a (AU) a (AU) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='23+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='28 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='217+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 30 60 90 120 150 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='28 e 30 60 90 120 150 i ( ) i ( ) = 78+27 25 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='99: Corner plot of HD 58368 Mpri (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='91+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='57+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) a (AU) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='031+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 Mpri (M ) 50 75 100 125 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 Msec (M ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 e 50 75 100 125 i ( ) i ( ) = 93+18 20 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='100: Corner plot of HD 207585 Article number, page 41 of 49 DAO 8上 HIPPARCOS 8E HIPPARCOS GAIA GAIA 8000 E 6E 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 站 6000F 4 4 4000E μα* (mas/yr) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 (mas/yr) 2 (s/wu) 2 2000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Mcomp(Mo) 0 o F 2 2000 2 A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 一 4000 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 6F 6000F 6 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1000 O-C O-C 0 0E 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1 1980 1982 1984 1986 1988 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)HIPPARCOS HIPPARCOS GAIA GAIA 10000 20F 30F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 5000 μα* (mas/yr) 30 (mas/yr) (s/u) 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0 40 10 5000 50 OF CORAVEL 10000 HERMES SALT-HRS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 60 2500 F 10E 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0F OF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2500 E 1990 1995 2000 2005 2010 2015 2020 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 20102015 Epoch (yr) Epoch (year) Epoch (year)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='101: RV curve and proper motions of HD 44896 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='102: RV curve and proper motions of HD 199939 Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='99+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 Msec (M ) Msec (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 a (AU) a (AU) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='288+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='060 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='019+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='012 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Mpri (M ) 50 75 100 125 150 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 Msec (M ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='95 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='55 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='060 e 50 75 100 125 150 i ( ) i ( ) = 78+35 21 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='103: Corner plot of HD 44896 Mpri (M ) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='97+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='73+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 a (AU) a (AU) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='281+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='012 2 3 4 5 6 Mpri (M ) 60 80 100 120 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Msec (M ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 e 60 80 100 120 i ( ) i ( ) = 82+21 13 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='104: Corner plot of HD 199939 Article number, page 42 of 49 10000 E 8F HIPPARCOS HIPPARCOS GAIA GAIA 7500E 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 5000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 6 μα* (mas/yr) 2500E (mas/yr) RV (m/s) 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 8 0 2500 E 10 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 5000 12 7500 E 4F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 14F 6 10000 CORAVEL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5 E 500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 O-C 0 0一 500F 5E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 21 1986 1988 1990 1992 1994 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)HIPPARCOS 10000 HIPPARCOS 16 GAIA GAIA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 7500E 18 0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 5000 (mas/yr) (μk/sew) RV (m/s) 2 20 2500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 22 4 n 2500 24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 5000 DAO 8上 HERMES 26 SOPHIE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 7500E L 2000F 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0 TMMMMNNMMMMMM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2000 E 2E 2000 1980 1990 2010 2020 1990 1995 20002005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='105: RV curve and proper motions of HD 123585 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='106: RV curve and proper motions of HD 24035 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 a (AU) a (AU) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='41+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='025+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Mpri (M ) 60 80 100 120 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 e 60 80 100 120 i ( ) i ( ) = 90+19 19 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='107: Corner plot of HD 123585 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='82+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='68 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='76+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 a (AU) a (AU) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='41+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='014+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='010 1 2 3 4 Mpri (M ) 30 60 90 120 150 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='08 e 30 60 90 120 150 i ( ) i ( ) = 71+31 22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='108: Corner plot of HD 24035 Article number, page 43 of 49 HIPPARCOS HIPPARCOS 10 GAIA GAIA 50 10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 15 45 20 5000 μα* (mas/yr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 (mas/yr) 40 RV (m/s) 25 35 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 Mcomp(Mo) oF 30 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 5000 F 25 E 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 20 E 45F 10000 ΦCORAVEL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 5 10日 1000 O-C OF 0 0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 1000 5 1990 1995 2000 20102015 1990 1995 2000 2005 1991 1992 1993 1994 1995 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)HIPPARCOS HIPPARCOS 10000 GAIA GAIA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 45 0 5 40 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 5000F (u/sew) (mas/yr) 10 (s/w) 35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 of 15 30 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 on 20 5000 25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 25 20 —10000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 CORAVEL 30 500F 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 O-C O-C O-C 0 0 500M 5 2 1986 1988 1990 1992 1994 1990 1995 2000 2005 2010 2015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='109: RV curve and proper motions of HD 224621 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='110: RV curve and proper motions of HD 87080.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' We used a fixed RV offset of 248 m/s (Escorza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Mpri (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='66+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 a (AU) a (AU) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='029+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='045 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='020+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 Mpri (M ) 30 60 90 120 150 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='045 e 30 60 90 120 150 i ( ) i ( ) = 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='111: Corner plot of HD 224621 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='36+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='70+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 a (AU) a (AU) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='054+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='200 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='162+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Mpri (M ) 50 75 100 125 150 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='200 e 50 75 100 125 150 i ( ) i ( ) = 60+11 12 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='112: Corner plot of HD 87080 Article number, page 44 of 49 10000 20 HIPPARCOS HIPPARCOS GAIA GAIA 7500 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 80 0 5000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 100 (mas/yr) (uk/sew) 2500 (s/u) 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 120 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 W 40 comp(M。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=') 2500 140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 5000 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 LB91 160 亚 7500 CES CORAVEL 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 25 10 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0 T O-C 0 0 1000 25 10 1990 1980 1985 1995 1990 1995 2000200520102015 1990 199520002005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)90 HIPPARCOS HIPPARCOS GAIA GAIA 10000 50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 55 80 5000 μα* (mas/yr) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 (u/sew) 60 RV (m/s) 70 0 65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 Mcomp(Mo) 70 5000 60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 75 10000H 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 CORAVEL 50 80 HERMES 15000 [ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 500 5 O-C 0-0 0-0 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 500 5 2010 20102015 1990 1995 2000 2005 2015 1990 1995 2000 2005 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='113: RV curve and proper motions of HD 121447 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='114: RV curve and proper motions of HD 77247 Mpri (M ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='59+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) a (AU) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='848+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='045 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0121+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 Mpri (M ) 30 60 90 120 150 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 Msec (M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='045 e 30 60 90 120 150 i ( ) i ( ) = 59+76 24 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='115: Corner plot of HD 121447 Mpri (M ) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='61+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='541+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='68 a (AU) a (AU) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='632+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='120 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1080+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0049 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 Mpri (M ) 50 75 100 125 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 Msec (M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='68 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='120 e 50 75 100 125 i ( ) i ( ) = 98+25 22 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='116: Corner plot of HD 77247 Article number, page 45 of 49 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 HIPPARCOS HIPPARCOS 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 GAIA GAIA 10000F 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 5000 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 (uk/sew) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 (uk/sew) (s/w) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 of 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4W 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 5000上: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 10000 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0[ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 CORAVEL 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 oF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0 0 1000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 6 1986 199019952000 2005 20102015 19901995 20002005 2010 2015 1988 1990 1992 1994 Epoch (yr) Epoch (year) Epoch (year)8F HIPPARCOS HIPPARCOS 4 10000 GAIA GAIA 6 7500 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 5000 8 μα* (mas/yr) (s/w) 2500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 10 2 0 12 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 2500 14 5000 2 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 75008 18 1988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 O-C O-C O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2500 1980 1982 1984 2010 2011 1990 1995 2000 200520102015 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Appendix B: Corner plots of HD 218356 Article number, page 46 of 49 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Mpri (M ) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='128+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) a (AU) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='785+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='072+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='045 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Mpri (M ) 40 80 120 160 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='45 Msec (M ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='20 e 40 80 120 160 i ( ) i ( ) = 90+42 41 Mpri (M ) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) Msec (M ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='85+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='18 18 24 30 36 a (AU) a (AU) = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 e e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='39+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='12 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Mpri (M ) 120 135 150 165 i ( ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 Msec (M ) 18 24 30 36 a (AU) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 e 120 135 150 165 i ( ) i ( ) = 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1: Corner plots of the inner (left) and outer (right) orbit of HD 218356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Article number, page 47 of 49 A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' main Appendix C: Two possible fits for HD 201657 Article number, page 48 of 49 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Escorza and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' De Rosa: Barium and related stars, and their white-dwarf companions III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' The masses of the white dwarfs Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1: Possible orbit for HD 201657 with a smaller eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Compatible with the orbit published by Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2: Possible orbit for HD 201657 with a larger eccentricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Twice the period published by Jorissen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='3: Corner plots associated with the fits for HD 201657 shown in figure C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='1 (left) and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content=' Article number, page 49 of 49 HIPPARCOS 36 GAIA 4000 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 34 E 2000 μα* (mas/yr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='9 10 μs (mas/yr) RV (m/s) 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='8 0 Mcomp(Mo) 8 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7 2000 6 26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 4 4000F CORAVEL HIPPARCOS 24 HERMES GAIA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5 1000 O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='4 O-C O-C 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='0 AA 0 1000 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='5E 1980 1990 2000 2010 2020 1990 1995 1990 1995 2000 2005 2010 2015 Epoch (yr) Epoch (year) Epoch (year)10000 30 F HIPPARCOS HIPPARCOS 55 GAIA O GAIA 8000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 50 [ 20 6000 45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 (uk/sew) (s/u) 4000 10F 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 2000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='25 2000 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='00 25 CORAVEL 10 4000F 不 HERMES 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='75 2000F O-C O-C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='50 2000 1980 1990 2000 2010 2020 1990 1995 2000 2005 20102015 1990 1995 2000 2005 20102015 Epoch (yr) Epoch (year) Epoch (year)Mpr (Mo) = 2.' metadata={'source': 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+page_content='878:3 a (AU) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='23+1:81 (AU) (nv) e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='159±0:06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='72-0:1 LODO 1000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='32 i()= 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='7: i(°) = 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EdE2T4oBgHgl3EQf9wni/content/2301.04232v1.pdf'} +page_content='6 160 75 C 心 0.' metadata={'source': 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turbines: Risk assessment from +larger-scale meteorology +Isabell Stucke1,2, Achim Zeileis1, Georg J. Mayr2, Thorsten Simon1, Gerhard Diendorfer3, +Wolfgang Schulz3, Hannes Pichler3, and Deborah Morgenstern1,2 +1Department of Statistics, University of Innsbruck, Austria +2Department of Atmospheric and Cryospheric Sciences, University of Innsbruck, Austria +3OVE Service GmbH, Dept. ALDIS (Austrian Lightning Detection & Information System), Vienna, Austria +Correspondence: Isabell Stucke (isabell.stucke@uibk.ac.at) +Abstract. Upward lightning has become an increasingly important threat to wind turbines as ever more of them are being +installed for renewably producing electricity. The taller the wind turbine the higher the risk that the type of lightning striking +the man-made structure is upward lightning. Upward lightning can be much more destructive than downward lightning due +to its long lasting initial continuous current leading to a large charge transfer within the lightning discharge process. Current +standards for the risk assessment of lightning at wind turbines mainly take the summer lightning activity into account, which +is inferred from LLS. Ground truth lightning current measurements reveal that less than 50 % of upward lightning might be +detected by lightning location systems (LLS). This leads to a large underestimation of the proportion of LLS-non-detectable +upward lightning at wind turbines, which is the dominant lightning type in the cold season. This study aims to assess the risk of +LLS-detectable and LLS-non-detectable upward lightning at wind turbines using direct upward lightning measurements at the +Gaisberg Tower (Austria) and Säntis Tower (Switzerland). Direct upward lightning observations are linked to meteorological +reanalysis data and joined by random forests, a powerful machine learning technique. The meteorological drivers for the +non-/occurrence of LLS-detectable and LLS-non-detectable upward lightning, respectively, are found from the random forest +models trained at the towers and have large predictive skill on independent data. In a second step the results from the tower- +trained models are extended to a larger study domain (Central and Northern Germany). The tower-trained models for LLS- +detectable lightning is independently verified at wind turbine locations in that domain and found to reliably diagnose that +type of upward lightning. Risk maps based on case study events show that high diagnosed probabilities in the study domain +coincide with actual upward lightning events. This lends credence to the transfer of the model for all upward lightning types, +which increases both the risk and the affected areas. +1 +Introduction +The growing importance to produce renewable energy has recently led to a notable increase in the number of wind turbines +(e.g., Pineda et al., 2018). Since those structures are commonly taller than 100 m, the initiation of upward lightning (UL) +propagating from the tall structure towards the clouds is facilitated (Berger, 1967). A tall structure is more prone to experience +1 +arXiv:2301.03360v1 [stat.ML] 9 Jan 2023 + +UL as it is exposed to a stronger electrical field in comparison to the ground. Structures shorter than 100 m mainly experience +downward lightning (DL) with leaders propagating from the clouds towards the earth surface (e.g., Rakov and Uman, 2003). +As wind turbines are getting taller, UL is the major weather-related cause of severe damages to them (e.g., Rachidi et al., +2008; Montanyà Puig et al., 2016; Pineda et al., 2018; Matsui et al., 2020; Zhang and Zhang, 2020). It can be much more +destructive than DL due to its initial continuous current (ICC) lasting approximately ten times longer than the current of DL. +Ground truth lightning current measurements at the specially instrumented tower on top of the Gaisberg mountain (Austria, +Salzburg) reveal that more than 50 % of UL is not detected by conventional lightning location systems (LLS). The reason is +that the LLS cannot detect a particular subtype of UL having only an ICC (Diendorfer et al., 2015; March et al., 2016). Even +though towers exist providing ground truth lightning current data for LLS-detectable UL (UL-LLS) such as the Säntis Tower +in Switzerland, the Gaisberg Tower is the only instrumented tower in Europe providing the full information on the occurrence +of both UL-LLS and LLS-non-detectable UL (UL-noLLS). +Standards for lightning protection of wind turbines (e.g., IEC61400-24, 2019) crucially underestimate the occurrence of UL +at wind turbines since they currently rely only on three factors: The height of the wind turbine, the local annual flash density +derived from LLS and an environmental term involving factors like terrain complexity or altitude (Rachidi et al., 2008; Pineda +et al., 2018). Lightning activity in summer clearly dominates the annual local flash density due to large amounts of DL caused +by deep convection. However, UL is expected to be the dominant lightning type at wind turbines with a tendency to be even +more important in the colder season (Diendorfer, 2020; Rachidi et al., 2008). Further the risk assessment standards cannot +account for UL-noLLS, but can only account for UL-LLS given that a tall structure is present. +The major objective of this study is to assess the risk of UL-LLS and UL-noLLS at wind turbines over a larger domain. Only +at very few points the actual occurrence of UL can be analyzed based on direct measurements. Even though LLS networks exist +which might allow to analyze UL-LLS at tall structures, the lightning current measurements show that a significant proportion +is missed. Being aware that conventional LLS cannot assess the full risk of UL at wind turbines, this study uses a new approach. +It uses machine learning techniques linking the occurrence of UL to the larger-scale meteorological setting. The occurrence +of UL can only be provided by ground truth lightning current measurements. These are the basis to build and train the statistical +models used to eventually assess the risk of UL over a whole study domain. Specifically, this study employs conditional +inference random forests (Hothorn and Zeileis, 2015), which account for highly nonlinear and complex interactions between +the incidence of UL to the tall structures and the atmosphere. The achievement of the major objective requires several steps. +From lightning current measurement data at two instrumented towers in Austria (Gaisberg Tower) and Switzerland (Säntis +Tower) two models are constructed: One for UL-LLS and one for UL-LLS + UL-noLLS. These shall first find whether there +is a relationship between larger-scale meteorological variables and the occurrence of UL and second demonstrate how well +larger-scale meteorology can serve as a diagnostic tool to infer the occurrence of UL. +The benefit of the availability of UL-LLS data helps to verify whether the results from the instrumented towers are trans- +ferable. The idea is to extract wind turbine locations within the study domain and identify all lightning strikes to them from +the colder season (ONDJFMA) using LLS data. Succeeding in reliably diagnosing UL-LLS from larger-scale meteorology in +2 + +combination with UL ground truth lightning current measurements provides a stronger reliability of the results when in a final +step the risk of UL-noLLS, which cannot be verified using LLS data, is assessed. +The following sections are organized as follows. Section 2 introduces the two instrumented towers providing the necessary +ground truth data for this study. The first one is the Gaisberg Tower providing both UL-LLS and UL-noLLS and the second +one is the Säntis Tower providing only UL-LLS. Further this section introduces the identification of lightning at wind turbines +in the study domain as well as the meteorological data used. +Section 3 summarizes the procedures and major findings from the two instrumented towers. Section 3.1 describes the basic +principle of the construction of a random forest model. Section 3.2 presents the performance of the models at the instrumented +towers. Further, the most important larger-scale meteorological variables are introduced which lead to a higher risk of UL +(Sect. 3.3). +Then, Sect. 4 presents the results extending the models from the instrumented towers to the larger study domain to find +regions with a higher risk to experience UL. Section 4.1 diagnoses UL-LLS at wind turbines and presents case studies. Then, +in Sect. 4.2 the risk of UL-LLS and UL-LLS + UL-noLLS at wind turbines is illustrated and discussed using the whole period +of consideration. +Section 5 concludes and summarizes the most important findings. +2 +Data +This study combines five different data sources: UL data measured directly at the Gaisberg Tower in Austria (Diendorfer et al., +2009) and at the Säntis Tower in Switzerland (Romero et al., 2012); LLS data measured remotely by the European Cooperation +for Lightning Detection (EUCLID, Schulz et al., 2016); larger-scale meteorological variables from the reanalysis database +ERA5 (Hersbach et al., 2020); wind turbine locations identified using the OpenStreetMap database. +2.1 +Direct UL measurements at instrumented towers +Figure 1 shows two of the very few instrumented towers for the direct measurement of currents from UL. These are the +Gaisberg Tower (1 288 m amsl, 47◦48′ N, 13◦60′ E) and the Säntis Tower (2 502 m amsl, 47◦14′ N, 9◦20′ E). Lightning at the +instrumented towers is almost exclusively UL. Gaisberg Tower recorded in total 819 UL events between 2000 and 2015. Säntis +Tower recorded 692 UL events between 2010 and 2017. +A sensitive shunt type sensor at Gaisberg allows to measure all types of upward flashes regardless of the current waveform, +i.e., UL-LLS and UL-noLLS. However, inductive sensors employed at Säntis cannot measure upward flashes with only an ICC +(approximately 50 %, Diendorfer et al., 2015). +Direct UL current measurements are the crucial prerequisite to construct the random forest models, which are extended to +the larger study domain after being trained on the tower data. The combination of data from both towers allows to construct +the two types of models, that shall diagnose UL-LLS and both UL-LLS + UL-noLLS. +3 + +2.2 +UL-LLS at wind turbines and study domain +Remotely detected lightning data by the LLS EUCLID and wind turbine locations derived from OpenStreetMap serve as +verification of the statistical models assessing the risk of UL-LLS for the selected study domain. +Within the study domain of 50°N–54°N and 6°E–16°E, 27 814 wind turbines have been installed by the end of 2020 (Fig. 1). +Having extracted the exact locations of these wind turbines, lightning strikes within a 0.003° circular area (approximately within +300 m radius) detected by EUCLID are identified and assumed as UL. EUCLID measures DL with a high flash detection +efficiency of more than 90 % (Schulz et al., 2016). As mentioned, UL might be detected less efficiently (< 50 % Diendorfer +et al., 2015). +Due to its destructive potential and its severe underestimation in the current lightning protection standards, UL shall be +explicitly accounted for investigating the risk of UL at wind turbines in the study domain. The tower-trained models are based +on UL data throughout the year. However, as UL is dominant in the colder season compared to DL, only the months from +October to April, starting from October 2018 until December 2020 are considered in the verification part of the study. Further, +since DL is dominant in the warmer season, the extraction of lightning strikes to wind turbines would possibly lead to ambiguity +in the identification of DL or UL when considering the whole year. +Figure 1. Geographic overview of the instrumented tower locations (Gaisberg and Säntis) as well as the study domain (box). Green dots +are manually identified wind turbine locations based on © OpenStreetMap 2020. Right: topographic map of study domain showing altitude +above mean sea level. Data taken from Shuttle Radar Topography Mission (Farr and Kobrick, 2000). +4 + +56°N +54°N +54°N +53°N +POL +NLD +52°N +52°N +Nordrh +50°N +51°N +CZE +48°N +isberg +50°N +6°E +8°E +10°E +12°E +14°E +16°E +Altitude +46°N +(m.a.m.s.I.) +0 +500 +1000 +5°E +10°E +15°E +ldentified wind turbine location2.3 +Meteorological data +ERA5 provides hourly reanalysis of the state of the atmosphere. It has a resolution of 31 km horizontally ( grid cell size +of 0.25 ° x 0.25 ° ) and 137 levels vertically. This study uses 35 directly available and derived variables at the surface, on +model levels and integrated vertically. These reflect variables relevant for cloud electrification, lightning and thunderstorms +(Morgenstern et al., 2022). A full list of variables can be found in the Appendix A. Data are spatially and temporally bilinearly +interpolated to each Gaisberg and Säntis Tower UL observation as well as to each grid cell within the study domain in the +verification part of this study. +3 +Methodological procedures and findings from the instrumented towers +This section provides the required background information on the basic methods as well as important outcomes from the +analysis at the instrumented Gaisberg Tower and Säntis Tower. Three different aspects shall be covered in the following: First +the principle how the basic model, i.e., a random forest, is constructed. Second, the performance of the models and third, which +variables are most important to identify favorable conditions for UL to occur or not. +3.1 +Construction and verification of the tower-trained random forests +A machine learning technique, which has been recently widely adopted in various scientific fields, is used to link larger-scale +meteorology and the occurrence of UL at the instrumented towers. Random forests are highly flexible and able to handle +nonlinear effects capturing complex interactions with respect to the stated modeling problem (Strobl et al., 2009). +The occurrence versus the non-occurrence of UL is a binary classification problem which is tackled using 35 larger-scale +meteorological variables (predictors). Each meteorological predictor is linked to a situation with or without UL at the Gaisberg +or Säntis Tower using a random forest. A random forest combines predictions from several decision trees, learned on randomly +chosen subsamples of the input data. +Specifically, the trees in the random forest are constructed by capturing the association between the binary response and +each of the predictor variables using permutation tests (also known as conditional inference, see Strasser and Weber (1999)). +The idea is that, in each step of the recursive tree construction, the one predictor variable is selected which has the highest +(most significant) association with the response variable. Then, the dataset is split with respect to this predictor variable in +order to separate the different response classes as well as possible. Splitting is repeated recursively in each of the subsets of the +data until a certain stopping criterion (e.g., regarding significance or subsample size) is met. The forest combines 500 of such +trees, where each tree is learned on randomly subsampled two-thirds of the full data set and only considering six randomly +selected predictors in each split. Finally, the random forest averages the predictions from the ensemble of trees, which stabilizes +and enhances the predictive performance. See Hothorn et al. (2006) and Hothorn and Zeileis (2015) for more details on the +algorithm and implementation. +5 + +To validate the resulting models, the input data are split into training and testing data samples. On the training data, the +models are trained and on the unseen testing data the diagnostic ability is assessed. Leave-one-out cross-validation is used for +validating the models for UL-LLS and UL-LLS + UL-noLLS. The first model for UL-LLS uses both Säntis data and Gaisberg +data to increase the size of the training data. The particular flash type that cannot be detected at the Säntis Tower is left out +from the Gaisberg data during the training procedure to ensure consistency. The second model for UL-LLS + UL-noLLS uses +only Gaisberg data, as only the Gaisberg Tower provides full information on all subtypes of UL. +Between 2000 and 2015, the Gaisberg Tower experienced 247 unique days with UL events. Between 2010 and 2017, the +Säntis Tower experienced 186 unique days. Combining UL days from both towers yields 406 unique days with UL. Each +training input data leaves out one of the 247 (406) days with UL to use it as test data. This is repeated until each of the 247 +(406) days has been left out once for training the random forest models. This results in 247 (406) different models trained on +equal numbers of situations with and without UL. +The input model response (i.e., did UL occur or not) is sampled such that the two classes are balanced, i.e., situations with +and without UL are present with equal proportions. Assessing the models’ performance, the models diagnose the conditional +probability on data not considered during training the models, i.e., on the respective day left out. We call the probability +conditional due to the balanced model response setup. To diagnose the conditional probability of UL on days without UL as +well, days without UL from each season are randomly sampled between 2000 and 2017. High diagnostic ability relates to high +probabilities whenever UL occurred at Gaisberg or Säntis in the particular situation (i.e., a high true positive rate) and low +probabilities whenever no UL occurred (i.e., a low false positive rate). +3.2 +Performance of the tower-trained random forests +The tower-trained random forest models can reliably diagnose both UL-LLS and UL-LLS + UL-noLLS when validated on +unseen withheld data from the towers. Figure 2 summarizes the cross-validated diagnostic ability according to the random +forests for UL-LLS + UL-noLLS (Gaisberg) and UL-LLS (Gaisberg + Säntis). Both model ensembles show a similarly good +diagnostic performance. The diagnosed median conditional probabilities are about 0.8 given that UL was observed in the +respective situation (minute). This indicates a high true positive rate. Similarly, for situations without lightning (right), the +conditional probabilities are low indicating a low false positive rate. +That the random forest including UL-noLLS has the highest diagnostic ability demonstrates that the proportion which cannot +be detected by conventional LLS can be indeed reliably diagnosed by larger-scale meteorology alone. This supports the idea +to also investigate the risk for unverifiable UL-noLLS and not only for UL-LLS. +3.3 +Meteorological drivers for UL-LLS at the instrumented towers +Random forests allow to assess the influence of individual variables on the models’ diagnostic performance. This is done by +computing the so-called permutation variable importance. The idea is to break up the relationship between the response variable +and one predictor variable by neglecting its information when assessing the models’ diagnostic performance. Neglecting the +information of one predictor variable is done by permutation, i.e., randomly shuffling its values and then assessing how much +6 + +0.00 +0.25 +0.50 +0.75 +1.00 +UL−LLS + +UL−noLLS +UL−LLS +No UL +Diagnosed conditional probability of UL +Figure 2. Distributions of diagnosed conditional probabilities in situations with or without UL events. Left: conditional UL probability +given that UL was observed in the particular minute (true positive) based on Gaisberg data including all subtypes of UL. Center: conditional +UL probability given that UL was observed in the particular minute based on Gaisberg and Säntis data combined. Right: conditional UL +probability on randomly sampled days without UL events (false positive). +the diagnostic performance decreases. Figure 3 visualizes the computed median permutation variable importance according to +100 different random forest models for UL-LLS. Each of the 100 models is based on a balanced proportion of situations with +UL and randomly chosen situations without UL. Results for UL-LLS and UL-LLS + UL-noLLS models are very similar. +Convective precipitation has the largest influence on the occurrence of UL according to the random forests based on direct +observations from the Gaisberg and the Säntis Tower (Fig. 3). Neglecting the information of this driver variable reduces the +diagnostic performance most. The second and third most important variables are the maximum updraft velocity and convective +available potential energy (CAPE), respectively. Statistically summarizing the three most important variables shows that CAPE +is both at the Säntis Tower and at the Gaisberg Tower rather low, when UL occurs (median value of 68 J kg−1). Convective +precipitation comes with a median value of 3.8 mm and maximum vertical updraft velocity with a median of − 1.5 m s−1. All +values are larger in magnitude than on "average" when considering every single hour in the considered time range. However, +in comparison to situations with deep convection, the order of magnitude is not exceptionally high as may be observed with +deep convection in which particularly the CAPE values are commonly higher than 500 J kg−1. An important reason for this +might be that at the instrumented towers, UL occurs approximately equally distributed throughout the year whereas intense +thunderstorms with deep convection and high CAPE values occur mainly in the summer season. Further this might suggest +that for UL to occur, requires a combination of many different processes interacting to form favorable conditions for UL which +might be even more complex than providing favorable conditions for deep convection. +Other important variables are the maximum precipitation rate, the vertical size of the thundercloud, the amount of ice crystals +and solid hydrometeors as well as the 2 m dewpoint temperature are influential. +7 + +Solid hydrometeors +(total column) +Ice crystals +(−20 °C to −40 °C) +Solid hydrometeors +(−20 °C to −40 °C) +2 m dewpoint +Ice crystals +(total column) +Cloud size +Max. precipitation +rate +CAPE +Maximum updraft +Convective precipitation +0.0 +0.1 +0.2 +0.3 +Variable importance +Figure 3. Median permutation variable importance according to 100 different random forests based on balanced proportions of situations +with and without UL at the Gaisberg and Säntis Tower. +4 +UL at wind turbines +The extraction of wind turbine locations and identification of lightning strikes to them within 300 m in the colder season +(ONDJFMA) shows that there are regions within the study domain that experience UL more frequently than others (see Fig. 4). +Interestingly, regions which are more often affected by UL (panel (b), dark pink) coincide with regions with many wind +turbines. However, in general it can be observed that regions with a high number of wind turbines (panel (a), dark green) do +not necessarily coincide with a high number of UL as can be seen in the North-Eastern parts of the study domain, for instance. +The following sections present and discuss the results when extending the findings from the instrumented towers to the study +domain in which wind turbine locations are extracted and the lightning activity to them is analyzed. +4.1 +Diagnosing UL-LLS at wind turbines from larger-scale meteorological conditions +The random forest models for UL-LLS and UL-LLS + UL-noLLS based on data from the two instrumented towers identified +larger-scale meteorological variables which are most important distinguishing situations with and without UL. +Now, the tower-trained random forest models are applied to the larger study domain to assess the risk of UL at wind turbines. +Lightning measurements from LLS data shall verify the results at identified wind turbine locations. +The following results are based on a similar procedure as described in Sect. 3.2 except that each grid cell ( 31 km x 31 km ) +of the study domain is used as test data instead of the cross-validated data from the instrumented towers. +8 + +Figure 4. Panel (a): number of wind turbines per grid cell derived from © OpenStreetMap 2020 data. Panel (b): number of hours per grid +cell with lightning at wind turbines derived from EUCLID data. +In the following, the tower-trained random forest models are applied to each grid cell of the study domain. To increase the +robustness of the results, again 100 different random forest models based on observations from the Gaisberg and the Säntis +Tower are used to diagnose the conditional probability of UL on the selected case studies over the study domain. The results +in this section visualize the median conditional probabilities diagnosed by the random forest models. +Case studies: UL-LLS at wind turbines +To illustrate the diagnostic ability of the tower-trained random forests for UL-LLS on days with UL events, three different case +study days are selected out of colder seasons between 2018 and 2020 in the study domain. +9 + +54N +54N +53N +53N +口 +52N +52N +S +51N +51N +M +M +口 +50N +(a) +50N +(b) +6-E +8.E +10·E +12·E +14·E +16-E +6-E +8.E +10·E +12·E +14·E +16·E +Hours with lightning +Windturbines +20 +40 +60 +80>=100 +to wind turbine(s) +5 +10 +15 +20 +3035(a) +50°N +51°N +52°N +53°N +54°N + 6°E + 8°E +10°E +12°E +14°E +16°E +(b) +50°N +51°N +52°N +53°N +54°N + 6°E + 8°E +10°E +12°E +14°E +16°E +(c) +50°N +51°N +52°N +53°N +54°N + 6°E + 8°E +10°E +12°E +14°E +16°E +0.5 +1.0 +1.5 +2.0 +Conv. precip. +(mm) +(d) +50°N +51°N +52°N +53°N +54°N + 6°E + 8°E +10°E +12°E +14°E +16°E +0.5 +1.0 +1.5 +2.0 +Conv. precip. +(mm) +(e) +50°N +51°N +52°N +53°N +54°N + 6°E + 8°E +10°E +12°E +14°E +16°E +−3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 +Max. updraft +(Pa s−1) +(f) +50°N +51°N +52°N +53°N +54°N + 6°E + 8°E +10°E +12°E +14°E +16°E +−3.0 −2.5 −2.0 −1.5 −1.0 −0.5 0.0 +Max. updraft +(Pa s−1) +(g) +50°N +51°N +52°N +53°N +54°N + 6°E + 8°E +10°E +12°E +14°E +16°E +0 +50 +100 +150 +200 +CAPE (J kg−1) +(h) +50°N +51°N +52°N +53°N +54°N + 6°E + 8°E +10°E +12°E +14°E +16°E +0 +50 +100 +150 +CAPE (J kg−1) +Figure 5. Larger-scale meteorological setting on the 4th March 2019 over the study domain. Left column illustrates the setting at 13 UTC, +right column at 14 UTC. From upper to lower: spatial distributions of isolines of the 850 hPa temperature (in intervals of 1 K), convective +precipitation, the maximum large-scale updraft velocity (negative values is upward motion) and CAPE. Darker colors indicates higher +magnitude. Dark gray dots in all figures are flashes within the considered hour and ERA5 grid cell derived from LLS EUCLID data. +10 + +Figure 6. Median diagnosed conditional probability of UL according to 100 random forest models based on Gaisberg and Säntis Tower data +(red areas). Yellow symbols are flashes within the considered hour derived from EUCLID data. Gray shaded areas are grid cells without wind +turbines. +11 + +2019-03-04 13:00:00 +2019-03-04 14:00:00 +54.0.N= +54.0oN = +53.5oN +53.5oN +53.0oN +53.0oN +52.5oN +52.5oN +52.0oN +52.0oN +51.5oN +51.5.N +51.0N +51.0oN +50.5.N = +1 +1 +60E +8E +10.E +12E +14.E +16.E +60E +8E +10.E +12E +14.E +16.E +2020-02-11 01:00:00 +2020-02-11 21:00:00 +54.0oN = +54.0oN = +53.5N +53.0oN +53.0oN +52.5N +52.5N +52.0oN +52.0oN +51.5oN = +51.5N +X +51.0oN +51.00N +50.5oN ≤ +50.5oN ≤ +11 +60E +8.E +10E +12.E +140E +16E +60E +8.E +10E +12.E +140E +16.E +2020-02-17 14:00:00 +2020-02-17 15:00:00 +54.0oN = +54.0oN +53.5oN +53.0N +53.0oN +52.5。N +52.5oN +52.0oN +52.0oN +51.5oN +51.5oN +51.0N +51.0N +50.5N +6.E +8E +10.E +12.E +14E +16.E +6.E +8.E +10.E +12E +140E +16.E +UL probability +(conditional) +0.2 +0.4 +0.6 +0.0 +0.8 +1.0The selected case study days are characterized by typical weather situations for the colder seasons in the mid-latitudes. The +atmosphere in the transition seasons and in winter tends to be highly variable and influenced by the succession of cyclones and +anticyclones determining the meteorological setting (Perry, 1987). +In particular the development and progression of mid-latitude cyclones provides favorable conditions for so-called wind- +field thunderstorms (Morgenstern et al., 2022). This thunderstorm type is among others associated with strong updrafts, high +amounts of precipitation as well as low but present CAPE. +The first case study is considered in more detail regarding the drivers identified at the instrumented towers (Fig. 3). Figure 5 +illustrates the larger scale isotherm locations, the spatial distribution of convective precipitation, the maximum updraft velocity +and CAPE on the 4th March 2019 at 13 UTC and 14 UTC. LLS detected lightning events to the identified wind turbines within +the particular hour are indicated as dark gray dots. +The meteorological setting is determined by the passage of a cold front ahead of a trough around noon. Densely packed +isotherms at 850 hPa crossing Northern and Central Germany from West to East indicate the approximate location of the +cold front in panels (a) and (b). The cold front implies locally enhanced amounts of convective precipitation in (c) and (d), +strong updrafts indicated by large negative values in (e) and (f) and slightly increased but in general low CAPE in (g) and +(h) in comparison to deep convection in summer. All three variables show maximum increased values in slightly different +areas within the study domain induced by the cold front. Convective precipitation shows increased values along the cold front, +whereas the other two variables have locally more concentrated areas with maximum values (e.g., maximum updraft velocity +in North/Central Germany). +Figure 6 visualizes the diagnosed conditional probability by the random forest models in red colors for all three case study +days. Panels (a) and (b) show the results for the particular case study discussed in Fig. 5. The diagnosed pattern is a result of +combining the influence of the three driver variables. This suggests that not a single variable can be responsible for the resulting +probability map but it is rather an interaction of different influential variables yielding areas with increased risk to experience +UL. +The yellow symbols again show lightning strikes over the considered hour. Identified lightning events in yellow require a +wind turbine within a distance of maximum 300 m as described in Sect. 2. All other tall structures that might have experienced +UL are not considered in this figure. Therefore, the diagnosed probabilities do not depend on wind turbine locations meaning +that high probabilities might be diagnosed even though there is no wind turbine installed. Grid cells without any wind turbines +are shaded in gray. +All three case study days in Fig. 6 show that areas with increased diagnosed probability for UL to occur coincide well with +identified lightning events in the respective hour over the study domain. In all three case studies there is a clear separation +between areas with very low diagnosed risk and areas with very high diagnosed risk to experience UL. +On the 11th February 2020 shown in panel (c) and (d) of Fig. 6, the study domain is in strong westerly flow again associated +with locally increased convective precipitation, CAPE and strong updrafts (not shown here). On the 17th February 2020, +the study domain is crossed by a cold front in higher altitudes (above 500 hPa). Regardless of the different meteorological +12 + +situation, the conditions are again similar to the other case studies showing increased values in the three driver variables that +highly influence the diagnosed conditional probability. +4.2 +Risk assessment of UL at wind turbines +Identifying areas with increased risk of UL due to larger-scale meteorological conditions is a valuable step towards the risk +assessment of lightning at wind turbines. The case studies clearly demonstrate that observed lightning at wind turbines coincide +with areas of increased probability diagnosed by the random forest models. The following analysis considers all events within +the considered period of time in which lightning at wind turbines was identified. Not only the models for UL-LLS shall +provide a risk assessment, but now random forests for UL-LLS + UL-noLLS are additionally applied to the study domain and +the considered time period. +The considered study period including the transition seasons and winter from 2018 to 2020 counts in total 185 event days +with 1 027 single flash hours and 18 602 single flash events. These numbers shall be a measure to verify the resulting diagnosed +probabilities by the random forest models. Note that these numbers are the lower limit of actually occurred flashes. Considering +the uncertainty of manually identifying flashes at wind turbines as well as the uncertainty of detecting UL by the LLS suggests a +significantly larger number of actually occurred lightning events at wind turbines. Further, this verification approach exclusively +considers lightning at wind turbines and neglects all other tall structures such as radio towers in the study domain that might +be affected by UL. In the following, all days within the considered study period are taken as new data for the random forest +models to diagnose the conditional probabilities on hourly basis. +The objective is to identify regions that are more frequently affected by a higher risk of UL compared to other regions +according to the random forest models. For this purpose the number of hours in each ERA5 grid cell ( 0.25 ° x 0.25 ° ) that +exceeds the conditional probability threshold of 0.5 is counted. +Risk assessment of UL-LLS at wind turbines +Figure 7 (a) illustrates that there are regions in the considered study domain having a higher risk to experience UL-LLS more +often than other regions. The western and southwestern parts of the study domain have a considerably higher probability for +UL-LLS. This is also in agreement with panel (b) in Fig. 4 showing the actually observed hours in which at least one lightning +event to a wind turbine occurred within the respective grid cell. +Interestingly, areas with higher UL-LLS probabilities in Fig. 7 roughly coincide with regions of elevated topography in the +southern third of the domain (cf. Fig. 1). Possible explanations are an increased lightning-effective height (e.g., Shindo, 2018) +of the turbines and increased chances for thunderstorm formation through orographic lifting and thermally-induced breezes +(Kirshbaum et al., 2018). Sea breezes might also be an explanation for the higher probabilities in the northwesternmost, sea- +covered part of the domain. +The successful transfer of the UL-LLS model trained with meteorological data on direct tower measurements to a larger +region and its independent verification on wind turbines shows the potential of our approach to be able to produce regionally +13 + +varying risk maps, which might in turn lead to regionally varying (voluntary or enforced) lightning protection standards for +wind turbines. +Risk assessment of UL-LLS + UL-noLLS at wind turbines +The successful transfer of the tower-trained and verified UL-LLS model to a larger domain lends credence to taking the same +step with the tower-trained model for all upward lightning (UL-LLS and UL-noLLS) although no data exist for an independent +verification. +Panel (b) in Fig. 7 indicates that more flashes are expected when additionally accounting for the LLS-non detectable UL +flash type. The pattern of areas with increased risk to experience UL are similar even though some areas affected more often +are enlarged. From this it can be suggested that there are similar mechanisms that result from larger-scale meteorology leading +to the UL-LLS or UL-noLLS flash types. +The risk in regions with elevated topography in the southern part of the domain and in the coastal northwesternmost region +is most pronouncedly increased. +5 +Conclusions +Upward lightning (UL) initiating at tall structures such as wind turbines is much more destructive than downward lightning +(DL). Each UL flash starts with an initial continuous current (ICC) lasting about ten times longer than in DL transferring much +more charge to the tall structure. Further, direct upward lightning measurements suggest that less than 50 % of UL events can +be detected by most lightning location systems (LLS) since they are not able to spot UL with only an ICC. +UL directly measured at the instrumented tower at Gaisberg has little seasonal variation. However, current lightning pro- +tection standards are based on the annual flash density derived from LLS data which is clearly dominated by DL in the warm +season. UL-noLLS is completely neglected and UL in the cold season is highly underestimated. Basic knowledge about the +occurrence of UL is still incomplete impeding a proper risk assessment of UL at wind turbines. +The missing consideration of UL-noLLS and of the importance of the cold season for UL will therefore considerably under- +estimate the risk of UL to wind turbines. This study leverages rare direct UL measurements with larger-scale meteorological +data in a machine learning model in order to estimate the risk of all UL (UL-LLS and UL-noLLS) at wind turbines. +The first step constitutes training and validating two different random forest models based on long-term observations from +two specially instrumented towers. One model accounts only for UL-LLS and one model accounts for UL-LLS + UL-noLLS. +The model input data are direct UL measurements from the Gaisberg Tower (Austria, 2000-2015) and from the Säntis Tower +(Switzerland, 2010-2017). While the sensor at the Gaisberg Tower measures also UL-noLLS, the sensor at the Säntis Tower +misses most of them. +In a second step, the random forest models are extended to a larger study domain (50°N - 54°N and 6°E -16°E) to identify +areas with increased risk of UL in the colder season (ONDJFMA). As a verification, all lightning strikes at wind turbines in this +domain are extracted from LLS and OpenStreetMap data and compared to the diagnosed probabilities by the random forests. +14 + +Düsseldorf +Leipzig +Hamburg +Berlin +(a) +50°N +51°N +52°N +53°N +54°N + 6°E + 8°E +10°E +12°E +14°E +16°E +UL−LLS +Düsseldorf +Leipzig +Hamburg +Berlin +(b) +50°N +51°N +52°N +53°N +54°N + 6°E + 8°E +10°E +12°E +14°E +16°E +UL−LLS + UL−noLLS +250 +(2 %) +500 +(4 %) +750 +(6 %) +1000 +(8 %) +1250 +(10 %) +Hours +(Rel. proportion) +Figure 7. Panels (a) and (b): potential maps for UL in the colder season (ONDJFMA) from 2018 to 2020. Orange colors are median of +hours per grid cell exceeding conditional probabilities of 0.5 according to 100 random forest models. Panel (a) shows results according to +models based on Gaisberg and Säntis data combined. Panel (b) shows results according to models based on Gaisberg data also including the +UL-noLLS. Relative proportion of in total 12480 hours are given as reference. +Results show that UL can be reliably diagnosed by the tower-trained random forest models at the Gaisberg and Säntis Tower. +The larger-scale meteorological drivers are large amounts of (convective) precipitation, strong vertical updraft velocities and +slightly increased CAPE. Further, the vertical extent of the cloud as well as the amount of ice crystals and solid hydrometeors +are important variables. +The extension of the random forests to a larger domain shows that probability maps coincide with observed lightning strikes +at wind turbines. Extending models trained at the Gaisberg Tower including UL-noLLS flashes reveals that areas with increased +risk to experience UL are expected to experience UL even more often. +The western and southern part of the domain in North-West Germany with elevated topography and the coastal region in +its northwesternmost part are most at risk of UL at wind turbines. This study demonstrates that direct UL measurements at an +instrumented tower can be reliably modeled from larger-scale meteorological conditions in a machine learning model (random +forest). The study also proposes a novel way how the transfer of that model to a larger region can be justified by using UL-LLS +data at wind turbine locations. Consequently regionally detailed risk maps of UL at wind turbines can be produced. +15 + +Appendix A: Additional material +A1 +Data availability +ERA5 data are freely available for download at https://cds.climate.copernicus.eu Hersbach et al. (2020). EUCLID data and +direct observations from the Gaisberg Tower are available only on request. For more details contact Wolfgang Schulz. +A2 +Software +All calculations as well as setting up the final data sets, modeling and predicting were performed using R (R Core Team, +2021), using packages netCDF4 (Pierce, 2019), partykit (Hothorn and Zeileis, 2015), ggplot2 package (Wickham, 2016). +Retrieving the raw data and deriving further variables from ERA5 required using Python3 (Van Rossum and Drake, 2009) and +cdo (Schulzweida, 2019). +A3 +Risk assessment of UL at wind turbines using a higher probability threshold +In Sect. 4.2 the model results for the risk assessment of UL-LLS and UL-LLS + UL-noLLS are presented in the way that hours +are counted exceeding a conditional probability of 0.5. Figure A1 illustrates the risk assessment using a higher probability +threshold, namely 0.8. The number of hours exceeding this threshold is lower by about a factor of two in comparison to a +probability threshold of 0.5. However, the regional pattern is still similar with maxima West/South-West of the study domain. +16 + +Düsseldorf +Leipzig +Hamburg +Berlin +(a) +50°N +51°N +52°N +53°N +54°N + 6°E + 8°E +10°E +12°E +14°E +16°E +UL−LLS +Düsseldorf +Leipzig +Hamburg +Berlin +(b) +50°N +51°N +52°N +53°N +54°N + 6°E + 8°E +10°E +12°E +14°E +16°E +UL−LLS + UL−noLLS +100 +(0.8 %) +200 +(1.6 %) +300 +(2.4 %) +400 +(3.2 %) +500 +(4 %) +600 +(4.8 %) +Hours +(Rel. proportion) +Figure A1. Panels (a) and (b): maps for the potential of UL in the colder season (ONDJFMA) from 2018 to 2020. Orange colors are median +of hours per grid cell exceeding conditional probabilities of 0.8 according to 100 random forest models. Panel (a) shows results according to +models based on Gaisberg and Säntis data combined. Panel (b) shows results according to models based on Gaisberg data also including the +UL-noLLS. Relative proportions of in total 12480 hours are given as reference. +17 + +Table A1. Table of large-scale variables taken from ERA5 and variables derived from ERA5. The derived variables (indicated in italics) are +suggested to be potentially important in the charging process of a thundercloud or for the development of convection. +Large-scale variables +Unit +cloud base height above ground +m agl +convective precipitation +(rain + snow) +m +large scale precipitation +m +cloud size +m +maximum precipitation rate +(rain + snow) +kg m−2 s−1 +ice crystals (total column, tciw) +kg m−2 +Solid hydrometeors (total column, tcsw) +kg m−2 +supercooled liquid water +(total column, tcslw) +kg m−2 +water vapor (total column) +kg m−2 +vertical integral of divergence +of cloud frozen water flux +kg m−2 s−1 +vertical transport of liquids +around −10 °C +kg Pa s−1 +ice crystals +(−10 °C - −20 °C) +kg m−2 +ice crystals +(−20 °C - −40 °C) +kg m−2 +cloud water droplets +(−10 °C - −20 °C) +kg m−2 +solid hydrometeors +(−10 °C - −20 °C) +kg m−2 +solid hydrometeors +(−20 °C - −40 °C) +kg m−2 +solids (cswc + ciwc) +around −10 °C +kg m−2 +liquids (clwc + crwc) +around −10 °C +kg m−2 +2 m dew point temperature +K +18 + +mean vertically integrated +moisture convergence +kg m−2 s−1 +water vapor +(−10 °C - −20 °C) +kg m−2 +boundary layer height +m +surface latent heat flux +J m−2 +surface sensible heat flux +J m−2 +downward surface solar radiation +J m−2 +convective available +potential energy +J kg−1 +convective inhibition present +binary +mean sea level pressure +Pa +height of −10 °C isotherm +m agl +boundary layer dissipation +J m−2 +Maximum vertical updraft velocity +Pa s−1 +total cloud shear +m s−1 +wind speed at 10 m +m s−1 +wind direction at 10 m +◦ +shear between 10 m and cloud base +m s−1 +19 + +Acknowledgements. We acknowledge the funding of this work by the Austrian Research Promotion Agency (FFG), project no. 872656 +and Austrian Science Fund (FWF) grant no. P 31836. We thank the EMC Group of the Swiss Federal Institute of Technology (EPFL) for +providing the data of the Säntis Tower strikes. Finally we thank Siemens, the operator of BLIDS for providing EUCLID data. +Author contributions. Isabell Stucke did the investigation, wrote software, visualized the results and wrote the paper. Deborah Morgernstern, +Thorsten Simon and Isabell Stucke performed data curation, built the data set, and derived variables based on ERA5 data. Thorsten Simon +contributed with coding concepts. Georg J. Mayr provided support on the meteorological analysis, data organization and funding acquisition. +Achim Zeileis supervised the formal analysis and interpretation of the statistical methods. Achim Zeileis, Georg J. Mayr, and Thorsten Simon +are the project administrators and supervisors. All authors contributed to the conceptualization of this paper, discussed on the methodology, +evaluated the results, and commented on the paper. +Competing interests. 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L.: Python 3 Reference Manual, Python Documentation Manual Part 2, CreateSpace Independent Publishing +Platform, Scotts Valley, CA, accessed 2021-03-09, 2009. +Wickham, H.: ggplot2: Elegant Graphics for Data Analysis, Springer-Verlag New York, https://ggplot2.tidyverse.org, 2016. +Zhang, Y. and Zhang, X.: Statistic analysis of lightning transients on wind turbines, Journal of Renewable and Sustainable Energy, 12, +063 302, https://doi.org/10.1063/5.0031506, 2020. +22 + diff --git a/I9E1T4oBgHgl3EQfsAU1/content/tmp_files/load_file.txt b/I9E1T4oBgHgl3EQfsAU1/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..189e12e0e0c493738b04cde7c225eadd7bc5dbf3 --- /dev/null +++ b/I9E1T4oBgHgl3EQfsAU1/content/tmp_files/load_file.txt @@ -0,0 +1,833 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf,len=832 +page_content='Upward lightning at wind turbines: Risk assessment from larger-scale meteorology Isabell Stucke1,2, Achim Zeileis1, Georg J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Mayr2, Thorsten Simon1, Gerhard Diendorfer3, Wolfgang Schulz3, Hannes Pichler3, and Deborah Morgenstern1,2 1Department of Statistics, University of Innsbruck, Austria 2Department of Atmospheric and Cryospheric Sciences, University of Innsbruck, Austria 3OVE Service GmbH, Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' ALDIS (Austrian Lightning Detection & Information System), Vienna, Austria Correspondence: Isabell Stucke (isabell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='stucke@uibk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='at) Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Upward lightning has become an increasingly important threat to wind turbines as ever more of them are being installed for renewably producing electricity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The taller the wind turbine the higher the risk that the type of lightning striking the man-made structure is upward lightning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Upward lightning can be much more destructive than downward lightning due to its long lasting initial continuous current leading to a large charge transfer within the lightning discharge process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Current standards for the risk assessment of lightning at wind turbines mainly take the summer lightning activity into account, which is inferred from LLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Ground truth lightning current measurements reveal that less than 50 % of upward lightning might be detected by lightning location systems (LLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' This leads to a large underestimation of the proportion of LLS-non-detectable upward lightning at wind turbines, which is the dominant lightning type in the cold season.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' This study aims to assess the risk of LLS-detectable and LLS-non-detectable upward lightning at wind turbines using direct upward lightning measurements at the Gaisberg Tower (Austria) and Säntis Tower (Switzerland).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Direct upward lightning observations are linked to meteorological reanalysis data and joined by random forests, a powerful machine learning technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The meteorological drivers for the non-/occurrence of LLS-detectable and LLS-non-detectable upward lightning, respectively, are found from the random forest models trained at the towers and have large predictive skill on independent data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' In a second step the results from the tower- trained models are extended to a larger study domain (Central and Northern Germany).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The tower-trained models for LLS- detectable lightning is independently verified at wind turbine locations in that domain and found to reliably diagnose that type of upward lightning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Risk maps based on case study events show that high diagnosed probabilities in the study domain coincide with actual upward lightning events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' This lends credence to the transfer of the model for all upward lightning types, which increases both the risk and the affected areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 1 Introduction The growing importance to produce renewable energy has recently led to a notable increase in the number of wind turbines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', Pineda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Since those structures are commonly taller than 100 m, the initiation of upward lightning (UL) propagating from the tall structure towards the clouds is facilitated (Berger, 1967).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' A tall structure is more prone to experience 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='03360v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='ML] 9 Jan 2023 UL as it is exposed to a stronger electrical field in comparison to the ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Structures shorter than 100 m mainly experience downward lightning (DL) with leaders propagating from the clouds towards the earth surface (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', Rakov and Uman, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' As wind turbines are getting taller, UL is the major weather-related cause of severe damages to them (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', Rachidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Montanyà Puig et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Pineda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Matsui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Zhang and Zhang, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' It can be much more destructive than DL due to its initial continuous current (ICC) lasting approximately ten times longer than the current of DL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Ground truth lightning current measurements at the specially instrumented tower on top of the Gaisberg mountain (Austria, Salzburg) reveal that more than 50 % of UL is not detected by conventional lightning location systems (LLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The reason is that the LLS cannot detect a particular subtype of UL having only an ICC (Diendorfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' March et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Even though towers exist providing ground truth lightning current data for LLS-detectable UL (UL-LLS) such as the Säntis Tower in Switzerland, the Gaisberg Tower is the only instrumented tower in Europe providing the full information on the occurrence of both UL-LLS and LLS-non-detectable UL (UL-noLLS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Standards for lightning protection of wind turbines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', IEC61400-24, 2019) crucially underestimate the occurrence of UL at wind turbines since they currently rely only on three factors: The height of the wind turbine, the local annual flash density derived from LLS and an environmental term involving factors like terrain complexity or altitude (Rachidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Pineda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Lightning activity in summer clearly dominates the annual local flash density due to large amounts of DL caused by deep convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' However, UL is expected to be the dominant lightning type at wind turbines with a tendency to be even more important in the colder season (Diendorfer, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Rachidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Further the risk assessment standards cannot account for UL-noLLS, but can only account for UL-LLS given that a tall structure is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The major objective of this study is to assess the risk of UL-LLS and UL-noLLS at wind turbines over a larger domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Only at very few points the actual occurrence of UL can be analyzed based on direct measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Even though LLS networks exist which might allow to analyze UL-LLS at tall structures, the lightning current measurements show that a significant proportion is missed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Being aware that conventional LLS cannot assess the full risk of UL at wind turbines, this study uses a new approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' It uses machine learning techniques linking the occurrence of UL to the larger-scale meteorological setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The occurrence of UL can only be provided by ground truth lightning current measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' These are the basis to build and train the statistical models used to eventually assess the risk of UL over a whole study domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Specifically, this study employs conditional inference random forests (Hothorn and Zeileis, 2015), which account for highly nonlinear and complex interactions between the incidence of UL to the tall structures and the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The achievement of the major objective requires several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' From lightning current measurement data at two instrumented towers in Austria (Gaisberg Tower) and Switzerland (Säntis Tower) two models are constructed: One for UL-LLS and one for UL-LLS + UL-noLLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' These shall first find whether there is a relationship between larger-scale meteorological variables and the occurrence of UL and second demonstrate how well larger-scale meteorology can serve as a diagnostic tool to infer the occurrence of UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The benefit of the availability of UL-LLS data helps to verify whether the results from the instrumented towers are trans- ferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The idea is to extract wind turbine locations within the study domain and identify all lightning strikes to them from the colder season (ONDJFMA) using LLS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Succeeding in reliably diagnosing UL-LLS from larger-scale meteorology in 2 combination with UL ground truth lightning current measurements provides a stronger reliability of the results when in a final step the risk of UL-noLLS, which cannot be verified using LLS data, is assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The following sections are organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Section 2 introduces the two instrumented towers providing the necessary ground truth data for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The first one is the Gaisberg Tower providing both UL-LLS and UL-noLLS and the second one is the Säntis Tower providing only UL-LLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Further this section introduces the identification of lightning at wind turbines in the study domain as well as the meteorological data used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Section 3 summarizes the procedures and major findings from the two instrumented towers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='1 describes the basic principle of the construction of a random forest model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='2 presents the performance of the models at the instrumented towers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Further, the most important larger-scale meteorological variables are introduced which lead to a higher risk of UL (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Then, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 4 presents the results extending the models from the instrumented towers to the larger study domain to find regions with a higher risk to experience UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='1 diagnoses UL-LLS at wind turbines and presents case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Then, in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='2 the risk of UL-LLS and UL-LLS + UL-noLLS at wind turbines is illustrated and discussed using the whole period of consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Section 5 concludes and summarizes the most important findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 2 Data This study combines five different data sources: UL data measured directly at the Gaisberg Tower in Austria (Diendorfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2009) and at the Säntis Tower in Switzerland (Romero et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' LLS data measured remotely by the European Cooperation for Lightning Detection (EUCLID, Schulz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' larger-scale meteorological variables from the reanalysis database ERA5 (Hersbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' wind turbine locations identified using the OpenStreetMap database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='1 Direct UL measurements at instrumented towers Figure 1 shows two of the very few instrumented towers for the direct measurement of currents from UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' These are the Gaisberg Tower (1 288 m amsl, 47◦48′ N, 13◦60′ E) and the Säntis Tower (2 502 m amsl, 47◦14′ N, 9◦20′ E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Lightning at the instrumented towers is almost exclusively UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Gaisberg Tower recorded in total 819 UL events between 2000 and 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Säntis Tower recorded 692 UL events between 2010 and 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' A sensitive shunt type sensor at Gaisberg allows to measure all types of upward flashes regardless of the current waveform, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', UL-LLS and UL-noLLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' However, inductive sensors employed at Säntis cannot measure upward flashes with only an ICC (approximately 50 %, Diendorfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Direct UL current measurements are the crucial prerequisite to construct the random forest models, which are extended to the larger study domain after being trained on the tower data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The combination of data from both towers allows to construct the two types of models, that shall diagnose UL-LLS and both UL-LLS + UL-noLLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='2 UL-LLS at wind turbines and study domain Remotely detected lightning data by the LLS EUCLID and wind turbine locations derived from OpenStreetMap serve as verification of the statistical models assessing the risk of UL-LLS for the selected study domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Within the study domain of 50°N–54°N and 6°E–16°E, 27 814 wind turbines have been installed by the end of 2020 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Having extracted the exact locations of these wind turbines, lightning strikes within a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='003° circular area (approximately within 300 m radius) detected by EUCLID are identified and assumed as UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' EUCLID measures DL with a high flash detection efficiency of more than 90 % (Schulz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' As mentioned, UL might be detected less efficiently (< 50 % Diendorfer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Due to its destructive potential and its severe underestimation in the current lightning protection standards, UL shall be explicitly accounted for investigating the risk of UL at wind turbines in the study domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The tower-trained models are based on UL data throughout the year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' However, as UL is dominant in the colder season compared to DL, only the months from October to April, starting from October 2018 until December 2020 are considered in the verification part of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Further, since DL is dominant in the warmer season, the extraction of lightning strikes to wind turbines would possibly lead to ambiguity in the identification of DL or UL when considering the whole year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Geographic overview of the instrumented tower locations (Gaisberg and Säntis) as well as the study domain (box).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Green dots are manually identified wind turbine locations based on © OpenStreetMap 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Right: topographic map of study domain showing altitude above mean sea level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Data taken from Shuttle Radar Topography Mission (Farr and Kobrick, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 4 56°N 54°N 54°N 53°N POL NLD 52°N 52°N Nordrh 50°N 51°N CZE 48°N isberg 50°N 6°E 8°E 10°E 12°E 14°E 16°E Altitude 46°N (m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=') 0 500 1000 5°E 10°E 15°E ldentified wind turbine location2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='3 Meteorological data ERA5 provides hourly reanalysis of the state of the atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' It has a resolution of 31 km horizontally ( grid cell size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='25 ° x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='25 ° ) and 137 levels vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' This study uses 35 directly available and derived variables at the surface, on model levels and integrated vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' These reflect variables relevant for cloud electrification, lightning and thunderstorms (Morgenstern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' A full list of variables can be found in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Data are spatially and temporally bilinearly interpolated to each Gaisberg and Säntis Tower UL observation as well as to each grid cell within the study domain in the verification part of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 3 Methodological procedures and findings from the instrumented towers This section provides the required background information on the basic methods as well as important outcomes from the analysis at the instrumented Gaisberg Tower and Säntis Tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Three different aspects shall be covered in the following: First the principle how the basic model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', a random forest, is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Second, the performance of the models and third, which variables are most important to identify favorable conditions for UL to occur or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='1 Construction and verification of the tower-trained random forests A machine learning technique, which has been recently widely adopted in various scientific fields, is used to link larger-scale meteorology and the occurrence of UL at the instrumented towers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Random forests are highly flexible and able to handle nonlinear effects capturing complex interactions with respect to the stated modeling problem (Strobl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The occurrence versus the non-occurrence of UL is a binary classification problem which is tackled using 35 larger-scale meteorological variables (predictors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Each meteorological predictor is linked to a situation with or without UL at the Gaisberg or Säntis Tower using a random forest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' A random forest combines predictions from several decision trees, learned on randomly chosen subsamples of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Specifically, the trees in the random forest are constructed by capturing the association between the binary response and each of the predictor variables using permutation tests (also known as conditional inference, see Strasser and Weber (1999)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The idea is that, in each step of the recursive tree construction, the one predictor variable is selected which has the highest (most significant) association with the response variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Then, the dataset is split with respect to this predictor variable in order to separate the different response classes as well as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Splitting is repeated recursively in each of the subsets of the data until a certain stopping criterion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', regarding significance or subsample size) is met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The forest combines 500 of such trees, where each tree is learned on randomly subsampled two-thirds of the full data set and only considering six randomly selected predictors in each split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Finally, the random forest averages the predictions from the ensemble of trees, which stabilizes and enhances the predictive performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' See Hothorn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' (2006) and Hothorn and Zeileis (2015) for more details on the algorithm and implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 5 To validate the resulting models, the input data are split into training and testing data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' On the training data, the models are trained and on the unseen testing data the diagnostic ability is assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Leave-one-out cross-validation is used for validating the models for UL-LLS and UL-LLS + UL-noLLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The first model for UL-LLS uses both Säntis data and Gaisberg data to increase the size of the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The particular flash type that cannot be detected at the Säntis Tower is left out from the Gaisberg data during the training procedure to ensure consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The second model for UL-LLS + UL-noLLS uses only Gaisberg data, as only the Gaisberg Tower provides full information on all subtypes of UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Between 2000 and 2015, the Gaisberg Tower experienced 247 unique days with UL events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Between 2010 and 2017, the Säntis Tower experienced 186 unique days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Combining UL days from both towers yields 406 unique days with UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Each training input data leaves out one of the 247 (406) days with UL to use it as test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' This is repeated until each of the 247 (406) days has been left out once for training the random forest models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' This results in 247 (406) different models trained on equal numbers of situations with and without UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The input model response (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', did UL occur or not) is sampled such that the two classes are balanced, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', situations with and without UL are present with equal proportions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Assessing the models’ performance, the models diagnose the conditional probability on data not considered during training the models, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', on the respective day left out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' We call the probability conditional due to the balanced model response setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' To diagnose the conditional probability of UL on days without UL as well, days without UL from each season are randomly sampled between 2000 and 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' High diagnostic ability relates to high probabilities whenever UL occurred at Gaisberg or Säntis in the particular situation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', a high true positive rate) and low probabilities whenever no UL occurred (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', a low false positive rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='2 Performance of the tower-trained random forests The tower-trained random forest models can reliably diagnose both UL-LLS and UL-LLS + UL-noLLS when validated on unseen withheld data from the towers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Figure 2 summarizes the cross-validated diagnostic ability according to the random forests for UL-LLS + UL-noLLS (Gaisberg) and UL-LLS (Gaisberg + Säntis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Both model ensembles show a similarly good diagnostic performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The diagnosed median conditional probabilities are about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='8 given that UL was observed in the respective situation (minute).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' This indicates a high true positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Similarly, for situations without lightning (right), the conditional probabilities are low indicating a low false positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' That the random forest including UL-noLLS has the highest diagnostic ability demonstrates that the proportion which cannot be detected by conventional LLS can be indeed reliably diagnosed by larger-scale meteorology alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' This supports the idea to also investigate the risk for unverifiable UL-noLLS and not only for UL-LLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='3 Meteorological drivers for UL-LLS at the instrumented towers Random forests allow to assess the influence of individual variables on the models’ diagnostic performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' This is done by computing the so-called permutation variable importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The idea is to break up the relationship between the response variable and one predictor variable by neglecting its information when assessing the models’ diagnostic performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Neglecting the information of one predictor variable is done by permutation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', randomly shuffling its values and then assessing how much 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='00 UL−LLS + UL−noLLS UL−LLS No UL Diagnosed conditional probability of UL Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Distributions of diagnosed conditional probabilities in situations with or without UL events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Left: conditional UL probability given that UL was observed in the particular minute (true positive) based on Gaisberg data including all subtypes of UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Center: conditional UL probability given that UL was observed in the particular minute based on Gaisberg and Säntis data combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Right: conditional UL probability on randomly sampled days without UL events (false positive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' the diagnostic performance decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Figure 3 visualizes the computed median permutation variable importance according to 100 different random forest models for UL-LLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Each of the 100 models is based on a balanced proportion of situations with UL and randomly chosen situations without UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Results for UL-LLS and UL-LLS + UL-noLLS models are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Convective precipitation has the largest influence on the occurrence of UL according to the random forests based on direct observations from the Gaisberg and the Säntis Tower (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Neglecting the information of this driver variable reduces the diagnostic performance most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The second and third most important variables are the maximum updraft velocity and convective available potential energy (CAPE), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Statistically summarizing the three most important variables shows that CAPE is both at the Säntis Tower and at the Gaisberg Tower rather low, when UL occurs (median value of 68 J kg−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Convective precipitation comes with a median value of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='8 mm and maximum vertical updraft velocity with a median of − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5 m s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' All values are larger in magnitude than on "average" when considering every single hour in the considered time range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' However, in comparison to situations with deep convection, the order of magnitude is not exceptionally high as may be observed with deep convection in which particularly the CAPE values are commonly higher than 500 J kg−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' An important reason for this might be that at the instrumented towers, UL occurs approximately equally distributed throughout the year whereas intense thunderstorms with deep convection and high CAPE values occur mainly in the summer season.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Further this might suggest that for UL to occur, requires a combination of many different processes interacting to form favorable conditions for UL which might be even more complex than providing favorable conditions for deep convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Other important variables are the maximum precipitation rate, the vertical size of the thundercloud, the amount of ice crystals and solid hydrometeors as well as the 2 m dewpoint temperature are influential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 7 Solid hydrometeors (total column) Ice crystals (−20 °C to −40 °C) Solid hydrometeors (−20 °C to −40 °C) 2 m dewpoint Ice crystals (total column) Cloud size Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' precipitation rate CAPE Maximum updraft Convective precipitation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='3 Variable importance Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Median permutation variable importance according to 100 different random forests based on balanced proportions of situations with and without UL at the Gaisberg and Säntis Tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 4 UL at wind turbines The extraction of wind turbine locations and identification of lightning strikes to them within 300 m in the colder season (ONDJFMA) shows that there are regions within the study domain that experience UL more frequently than others (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Interestingly, regions which are more often affected by UL (panel (b), dark pink) coincide with regions with many wind turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' However, in general it can be observed that regions with a high number of wind turbines (panel (a), dark green) do not necessarily coincide with a high number of UL as can be seen in the North-Eastern parts of the study domain, for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The following sections present and discuss the results when extending the findings from the instrumented towers to the study domain in which wind turbine locations are extracted and the lightning activity to them is analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='1 Diagnosing UL-LLS at wind turbines from larger-scale meteorological conditions The random forest models for UL-LLS and UL-LLS + UL-noLLS based on data from the two instrumented towers identified larger-scale meteorological variables which are most important distinguishing situations with and without UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Now, the tower-trained random forest models are applied to the larger study domain to assess the risk of UL at wind turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Lightning measurements from LLS data shall verify the results at identified wind turbine locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The following results are based on a similar procedure as described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='2 except that each grid cell ( 31 km x 31 km ) of the study domain is used as test data instead of the cross-validated data from the instrumented towers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 8 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Panel (a): number of wind turbines per grid cell derived from © OpenStreetMap 2020 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Panel (b): number of hours per grid cell with lightning at wind turbines derived from EUCLID data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' In the following, the tower-trained random forest models are applied to each grid cell of the study domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' To increase the robustness of the results, again 100 different random forest models based on observations from the Gaisberg and the Säntis Tower are used to diagnose the conditional probability of UL on the selected case studies over the study domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The results in this section visualize the median conditional probabilities diagnosed by the random forest models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Case studies: UL-LLS at wind turbines To illustrate the diagnostic ability of the tower-trained random forests for UL-LLS on days with UL events, three different case study days are selected out of colder seasons between 2018 and 2020 in the study domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 9 54N 54N 53N 53N 口 52N 52N S 51N 51N M M 口 50N (a) 50N (b) 6-E 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 10·E 12·E 14·E 16-E 6-E 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 10·E 12·E 14·E 16·E Hours with lightning Windturbines 20 40 60 80>=100 to wind turbine(s) 5 10 15 20 3035(a) 50°N 51°N 52°N 53°N 54°N 6°E 8°E 10°E 12°E 14°E 16°E (b) 50°N 51°N 52°N 53°N 54°N 6°E 8°E 10°E 12°E 14°E 16°E (c) 50°N 51°N 52°N 53°N 54°N 6°E 8°E 10°E 12°E 14°E 16°E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' precip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' (mm) (d) 50°N 51°N 52°N 53°N 54°N 6°E 8°E 10°E 12°E 14°E 16°E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0 Conv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' precip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' (mm) (e) 50°N 51°N 52°N 53°N 54°N 6°E 8°E 10°E 12°E 14°E 16°E −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' updraft (Pa s−1) (f) 50°N 51°N 52°N 53°N 54°N 6°E 8°E 10°E 12°E 14°E 16°E −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0 Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' updraft (Pa s−1) (g) 50°N 51°N 52°N 53°N 54°N 6°E 8°E 10°E 12°E 14°E 16°E 0 50 100 150 200 CAPE (J kg−1) (h) 50°N 51°N 52°N 53°N 54°N 6°E 8°E 10°E 12°E 14°E 16°E 0 50 100 150 CAPE (J kg−1) Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Larger-scale meteorological setting on the 4th March 2019 over the study domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Left column illustrates the setting at 13 UTC, right column at 14 UTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' From upper to lower: spatial distributions of isolines of the 850 hPa temperature (in intervals of 1 K), convective precipitation, the maximum large-scale updraft velocity (negative values is upward motion) and CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Darker colors indicates higher magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Dark gray dots in all figures are flashes within the considered hour and ERA5 grid cell derived from LLS EUCLID data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 10 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Median diagnosed conditional probability of UL according to 100 random forest models based on Gaisberg and Säntis Tower data (red areas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Yellow symbols are flashes within the considered hour derived from EUCLID data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Gray shaded areas are grid cells without wind turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 11 2019-03-04 13:00:00 2019-03-04 14:00:00 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='N= 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5oN 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5oN 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5oN 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5oN 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5oN 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='N 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0N 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='N = 1 1 60E 8E 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 12E 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 60E 8E 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 12E 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 2020-02-11 01:00:00 2020-02-11 21:00:00 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN = 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5N 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5N 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5N 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5oN = 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5N X 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='00N 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5oN ≤ 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5oN ≤ 11 60E 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 10E 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 140E 16E 60E 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 10E 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 140E 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 2020-02-17 14:00:00 2020-02-17 15:00:00 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN = 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5oN 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0N 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='N 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5oN 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0oN 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5oN 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5oN 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0N 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0N 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5N 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 8E 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 14E 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E 12E 140E 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='E UL probability (conditional) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='0The selected case study days are characterized by typical weather situations for the colder seasons in the mid-latitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The atmosphere in the transition seasons and in winter tends to be highly variable and influenced by the succession of cyclones and anticyclones determining the meteorological setting (Perry, 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' In particular the development and progression of mid-latitude cyclones provides favorable conditions for so-called wind- field thunderstorms (Morgenstern et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' This thunderstorm type is among others associated with strong updrafts, high amounts of precipitation as well as low but present CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The first case study is considered in more detail regarding the drivers identified at the instrumented towers (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Figure 5 illustrates the larger scale isotherm locations, the spatial distribution of convective precipitation, the maximum updraft velocity and CAPE on the 4th March 2019 at 13 UTC and 14 UTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' LLS detected lightning events to the identified wind turbines within the particular hour are indicated as dark gray dots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The meteorological setting is determined by the passage of a cold front ahead of a trough around noon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Densely packed isotherms at 850 hPa crossing Northern and Central Germany from West to East indicate the approximate location of the cold front in panels (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The cold front implies locally enhanced amounts of convective precipitation in (c) and (d), strong updrafts indicated by large negative values in (e) and (f) and slightly increased but in general low CAPE in (g) and (h) in comparison to deep convection in summer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' All three variables show maximum increased values in slightly different areas within the study domain induced by the cold front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Convective precipitation shows increased values along the cold front, whereas the other two variables have locally more concentrated areas with maximum values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', maximum updraft velocity in North/Central Germany).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Figure 6 visualizes the diagnosed conditional probability by the random forest models in red colors for all three case study days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Panels (a) and (b) show the results for the particular case study discussed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The diagnosed pattern is a result of combining the influence of the three driver variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' This suggests that not a single variable can be responsible for the resulting probability map but it is rather an interaction of different influential variables yielding areas with increased risk to experience UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The yellow symbols again show lightning strikes over the considered hour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Identified lightning events in yellow require a wind turbine within a distance of maximum 300 m as described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' All other tall structures that might have experienced UL are not considered in this figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Therefore, the diagnosed probabilities do not depend on wind turbine locations meaning that high probabilities might be diagnosed even though there is no wind turbine installed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Grid cells without any wind turbines are shaded in gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' All three case study days in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 6 show that areas with increased diagnosed probability for UL to occur coincide well with identified lightning events in the respective hour over the study domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' In all three case studies there is a clear separation between areas with very low diagnosed risk and areas with very high diagnosed risk to experience UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' On the 11th February 2020 shown in panel (c) and (d) of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 6, the study domain is in strong westerly flow again associated with locally increased convective precipitation, CAPE and strong updrafts (not shown here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' On the 17th February 2020, the study domain is crossed by a cold front in higher altitudes (above 500 hPa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Regardless of the different meteorological 12 situation, the conditions are again similar to the other case studies showing increased values in the three driver variables that highly influence the diagnosed conditional probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='2 Risk assessment of UL at wind turbines Identifying areas with increased risk of UL due to larger-scale meteorological conditions is a valuable step towards the risk assessment of lightning at wind turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The case studies clearly demonstrate that observed lightning at wind turbines coincide with areas of increased probability diagnosed by the random forest models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The following analysis considers all events within the considered period of time in which lightning at wind turbines was identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Not only the models for UL-LLS shall provide a risk assessment, but now random forests for UL-LLS + UL-noLLS are additionally applied to the study domain and the considered time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The considered study period including the transition seasons and winter from 2018 to 2020 counts in total 185 event days with 1 027 single flash hours and 18 602 single flash events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' These numbers shall be a measure to verify the resulting diagnosed probabilities by the random forest models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Note that these numbers are the lower limit of actually occurred flashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Considering the uncertainty of manually identifying flashes at wind turbines as well as the uncertainty of detecting UL by the LLS suggests a significantly larger number of actually occurred lightning events at wind turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Further, this verification approach exclusively considers lightning at wind turbines and neglects all other tall structures such as radio towers in the study domain that might be affected by UL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' In the following, all days within the considered study period are taken as new data for the random forest models to diagnose the conditional probabilities on hourly basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The objective is to identify regions that are more frequently affected by a higher risk of UL compared to other regions according to the random forest models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' For this purpose the number of hours in each ERA5 grid cell ( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='25 ° x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='25 ° ) that exceeds the conditional probability threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5 is counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Risk assessment of UL-LLS at wind turbines Figure 7 (a) illustrates that there are regions in the considered study domain having a higher risk to experience UL-LLS more often than other regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The western and southwestern parts of the study domain have a considerably higher probability for UL-LLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' This is also in agreement with panel (b) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 4 showing the actually observed hours in which at least one lightning event to a wind turbine occurred within the respective grid cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Interestingly, areas with higher UL-LLS probabilities in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 7 roughly coincide with regions of elevated topography in the southern third of the domain (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Possible explanations are an increased lightning-effective height (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', Shindo, 2018) of the turbines and increased chances for thunderstorm formation through orographic lifting and thermally-induced breezes (Kirshbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Sea breezes might also be an explanation for the higher probabilities in the northwesternmost, sea- covered part of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The successful transfer of the UL-LLS model trained with meteorological data on direct tower measurements to a larger region and its independent verification on wind turbines shows the potential of our approach to be able to produce regionally 13 varying risk maps, which might in turn lead to regionally varying (voluntary or enforced) lightning protection standards for wind turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Risk assessment of UL-LLS + UL-noLLS at wind turbines The successful transfer of the tower-trained and verified UL-LLS model to a larger domain lends credence to taking the same step with the tower-trained model for all upward lightning (UL-LLS and UL-noLLS) although no data exist for an independent verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Panel (b) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 7 indicates that more flashes are expected when additionally accounting for the LLS-non detectable UL flash type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The pattern of areas with increased risk to experience UL are similar even though some areas affected more often are enlarged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' From this it can be suggested that there are similar mechanisms that result from larger-scale meteorology leading to the UL-LLS or UL-noLLS flash types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The risk in regions with elevated topography in the southern part of the domain and in the coastal northwesternmost region is most pronouncedly increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 5 Conclusions Upward lightning (UL) initiating at tall structures such as wind turbines is much more destructive than downward lightning (DL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Each UL flash starts with an initial continuous current (ICC) lasting about ten times longer than in DL transferring much more charge to the tall structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Further, direct upward lightning measurements suggest that less than 50 % of UL events can be detected by most lightning location systems (LLS) since they are not able to spot UL with only an ICC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' UL directly measured at the instrumented tower at Gaisberg has little seasonal variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' However, current lightning pro- tection standards are based on the annual flash density derived from LLS data which is clearly dominated by DL in the warm season.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' UL-noLLS is completely neglected and UL in the cold season is highly underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Basic knowledge about the occurrence of UL is still incomplete impeding a proper risk assessment of UL at wind turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The missing consideration of UL-noLLS and of the importance of the cold season for UL will therefore considerably under- estimate the risk of UL to wind turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' This study leverages rare direct UL measurements with larger-scale meteorological data in a machine learning model in order to estimate the risk of all UL (UL-LLS and UL-noLLS) at wind turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The first step constitutes training and validating two different random forest models based on long-term observations from two specially instrumented towers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' One model accounts only for UL-LLS and one model accounts for UL-LLS + UL-noLLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The model input data are direct UL measurements from the Gaisberg Tower (Austria, 2000-2015) and from the Säntis Tower (Switzerland, 2010-2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' While the sensor at the Gaisberg Tower measures also UL-noLLS, the sensor at the Säntis Tower misses most of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' In a second step, the random forest models are extended to a larger study domain (50°N - 54°N and 6°E -16°E) to identify areas with increased risk of UL in the colder season (ONDJFMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' As a verification, all lightning strikes at wind turbines in this domain are extracted from LLS and OpenStreetMap data and compared to the diagnosed probabilities by the random forests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 14 Düsseldorf Leipzig Hamburg Berlin (a) 50°N 51°N 52°N 53°N 54°N 6°E 8°E 10°E 12°E 14°E 16°E UL−LLS Düsseldorf Leipzig Hamburg Berlin (b) 50°N 51°N 52°N 53°N 54°N 6°E 8°E 10°E 12°E 14°E 16°E UL−LLS + UL−noLLS 250 (2 %) 500 (4 %) 750 (6 %) 1000 (8 %) 1250 (10 %) Hours (Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' proportion) Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Panels (a) and (b): potential maps for UL in the colder season (ONDJFMA) from 2018 to 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Orange colors are median of hours per grid cell exceeding conditional probabilities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5 according to 100 random forest models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Panel (a) shows results according to models based on Gaisberg and Säntis data combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Panel (b) shows results according to models based on Gaisberg data also including the UL-noLLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Relative proportion of in total 12480 hours are given as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Results show that UL can be reliably diagnosed by the tower-trained random forest models at the Gaisberg and Säntis Tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The larger-scale meteorological drivers are large amounts of (convective) precipitation, strong vertical updraft velocities and slightly increased CAPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Further, the vertical extent of the cloud as well as the amount of ice crystals and solid hydrometeors are important variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The extension of the random forests to a larger domain shows that probability maps coincide with observed lightning strikes at wind turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Extending models trained at the Gaisberg Tower including UL-noLLS flashes reveals that areas with increased risk to experience UL are expected to experience UL even more often.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The western and southern part of the domain in North-West Germany with elevated topography and the coastal region in its northwesternmost part are most at risk of UL at wind turbines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' This study demonstrates that direct UL measurements at an instrumented tower can be reliably modeled from larger-scale meteorological conditions in a machine learning model (random forest).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The study also proposes a novel way how the transfer of that model to a larger region can be justified by using UL-LLS data at wind turbine locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Consequently regionally detailed risk maps of UL at wind turbines can be produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 15 Appendix A: Additional material A1 Data availability ERA5 data are freely available for download at https://cds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='climate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='copernicus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='eu Hersbach et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' EUCLID data and direct observations from the Gaisberg Tower are available only on request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' For more details contact Wolfgang Schulz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' A2 Software All calculations as well as setting up the final data sets, modeling and predicting were performed using R (R Core Team, 2021), using packages netCDF4 (Pierce, 2019), partykit (Hothorn and Zeileis, 2015), ggplot2 package (Wickham, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Retrieving the raw data and deriving further variables from ERA5 required using Python3 (Van Rossum and Drake, 2009) and cdo (Schulzweida, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' A3 Risk assessment of UL at wind turbines using a higher probability threshold In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='2 the model results for the risk assessment of UL-LLS and UL-LLS + UL-noLLS are presented in the way that hours are counted exceeding a conditional probability of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Figure A1 illustrates the risk assessment using a higher probability threshold, namely 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The number of hours exceeding this threshold is lower by about a factor of two in comparison to a probability threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' However, the regional pattern is still similar with maxima West/South-West of the study domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 16 Düsseldorf Leipzig Hamburg Berlin (a) 50°N 51°N 52°N 53°N 54°N 6°E 8°E 10°E 12°E 14°E 16°E UL−LLS Düsseldorf Leipzig Hamburg Berlin (b) 50°N 51°N 52°N 53°N 54°N 6°E 8°E 10°E 12°E 14°E 16°E UL−LLS + UL−noLLS 100 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='8 %) 200 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='6 %) 300 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='4 %) 400 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='2 %) 500 (4 %) 600 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='8 %) Hours (Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' proportion) Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Panels (a) and (b): maps for the potential of UL in the colder season (ONDJFMA) from 2018 to 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Orange colors are median of hours per grid cell exceeding conditional probabilities of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='8 according to 100 random forest models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Panel (a) shows results according to models based on Gaisberg and Säntis data combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Panel (b) shows results according to models based on Gaisberg data also including the UL-noLLS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Relative proportions of in total 12480 hours are given as reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 17 Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Table of large-scale variables taken from ERA5 and variables derived from ERA5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The derived variables (indicated in italics) are suggested to be potentially important in the charging process of a thundercloud or for the development of convection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Large-scale variables Unit cloud base height above ground m agl convective precipitation (rain + snow) m large scale precipitation m cloud size m maximum precipitation rate (rain + snow) kg m−2 s−1 ice crystals (total column,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' tciw) kg m−2 Solid hydrometeors (total column,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' tcsw) kg m−2 supercooled liquid water (total column,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' tcslw) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='kg m−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='water vapor (total column) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='kg m−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='vertical integral of divergence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='of cloud frozen water flux ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='kg m−2 s−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='vertical transport of liquids ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='around −10 °C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='kg Pa s−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='ice crystals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='(−10 °C - −20 °C) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='kg m−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='ice crystals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='(−20 °C - −40 °C) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='kg m−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='cloud water droplets ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='(−10 °C - −20 °C) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='kg m−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='solid hydrometeors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='(−10 °C - −20 °C) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='kg m−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='solid hydrometeors ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='(−20 °C - −40 °C) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='kg m−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='solids (cswc + ciwc) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='around −10 °C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='kg m−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content='liquids (clwc + crwc) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} 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+page_content='Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' We acknowledge the funding of this work by the Austrian Research Promotion Agency (FFG), project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 872656 and Austrian Science Fund (FWF) grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' P 31836.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' We thank the EMC Group of the Swiss Federal Institute of Technology (EPFL) for providing the data of the Säntis Tower strikes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Finally we thank Siemens, the operator of BLIDS for providing EUCLID data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Author contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Isabell Stucke did the investigation, wrote software, visualized the results and wrote the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Deborah Morgernstern, Thorsten Simon and Isabell Stucke performed data curation, built the data set, and derived variables based on ERA5 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Thorsten Simon contributed with coding concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Georg J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Mayr provided support on the meteorological analysis, data organization and funding acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Achim Zeileis supervised the formal analysis and interpretation of the statistical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Achim Zeileis, Georg J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Mayr, and Thorsten Simon are the project administrators and supervisors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' All authors contributed to the conceptualization of this paper, discussed on the methodology, evaluated the results, and commented on the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' Competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' The authors declare that they have no conflict of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=' 20 References Berger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9E1T4oBgHgl3EQfsAU1/content/2301.03360v1.pdf'} +page_content=': Novel observations on lightning discharges: Results of research on Mount San Salvatore, Journal of 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This study aims to solve the unsuper- +vised outlier detection problem where training +data contain outliers, but any label information +about inliers and outliers is not given. We pro- +pose a powerful and efficient learning framework +to identify outliers in a training data set using +deep neural networks. We start with a new obser- +vation called the inlier-memorization (IM) effect. +When we train a deep generative model with data +contaminated with outliers, the model first memo- +rizes inliers before outliers. Exploiting this find- +ing, we develop a new method called the outlier +detection via the IM effect (ODIM). The ODIM +only requires a few updates; thus, it is computa- +tionally efficient, tens of times faster than other +deep-learning-based algorithms. Also, the ODIM +filters out outliers successfully, regardless of the +types of data, such as tabular, image, and sequen- +tial. We empirically demonstrate the superiority +and efficiency of the ODIM by analyzing 20 data +sets. +1. Introduction +Outlier (also anomaly) is an observation that differs signifi- +cantly from other observations, and outlier detection (OD) +is the task of identifying outliers in a given data set. OD +has wide applications such as fraud detection, fault detec- +tion, and defect detection in images. OD is also used as +a pre-processing step in supervised learning to filter out +anomalous training samples, which may degrade the perfor- +mance of a predictive model. +OD problems can be categorized into three areas in general: +1) Supervised outlier detection (SOD) requires label infor- +1Department of Statistics, Sungshin Women’s University 2SK +Telecom 3Department of Statistics, Seoul National University. Cor- +respondence to: Yongdai Kim . +Preprint. +mation about whether each training sample is inlier (also +normal) or outlier and solves the two-class classification +task. A limitation of SOD is that it is hard to access the en- +tirely labeled data set in practice. 2) Semi-supervised outlier +detection (SSOD) refers to methods that assume all training +data being inliers and construct patterns or models for the +inliers. SSOD can be interpreted as the one-class classifica- +tion task since information of outliers is not used during the +training procedure. Similarly to SOD, it is not common to +have a data set composed of only inliers (Chandola et al., +2009; Chalapathy & Chawla, 2019). 3) Unsupervised out- +lier detection (UOD) deals with the most realistic situations +where training data include some outliers but no label in- +formation about anomalousness is available. Most anomaly +detection tasks in practice are related to UOD since the in- +formation of outliers in massive data is hardly known in +advance. +In this study, we propose a novel algorithm for UOD prob- +lems. Our algorithm is motivated by so called the memo- +rization effect observed in noisy label problems (Arpit et al., +2017; Jiang et al., 2018). The goal of noisy label problems is +to learn an accurate classifier when some of the class labels +in the training data are contaminated. When standard su- +pervised learning algorithms are applied to such mislabeled +data, an interesting phenomenon so called the memorization +effect is observed where correctly labeled data are learned +earlier and mislabeled data are learned later in the training +phase of deep neural networks. The memorization effect +makes it possible to detect mislabeled data by comparing +per-sample losses in an early stage of the training phase. +The aim of this paper is to apply the memorization effect to +UOD problems to develop a novel algorithm for detecting +outliers with high accuracy as well as high efficiency in +computation. +There already exists a study utilizing the memorization ef- +fect for UOD problems. (Wang et al., 2019a) noticed that +during a deep discriminative model is trained via the self- +supervised learning framework, the model memorizes in- +liers first and outliers next in the training phase, and named +this phenomenon the inlier-priority effect. Generating more +than a hundred artificial classes with a pre-specified anno- +tation strategy, they suggested a method, called E3-Outlier, +arXiv:2301.04257v1 [stat.ML] 11 Jan 2023 + +ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +which identifies outliers with high accuracy. +Even though it works effectively, but E3-Outlier is special- +ized to image data and not easy to be extended for other do- +mains of data such as tabular data, sequential data as well as +special image data including wafer images generated from +semiconductor fabrications. This is because E3-Outlier an- +notates the training data using a method that is specialized +for image data. +As a domain-agnostic UOD solver which can be used as an +off-the-shelf method, we develop a new method that inherits +the idea of the memorization effect but does not require any +prior expertise in data. We start with finding a new and +interesting observation that the memorization effect is also +observed in learning a deep generative model. That is, when +we train a deep generative model with training data that in- +cludes outliers, the inliers’ loss values reduce prior to those +of outliers at early updates. We call this observation the +inlier-memorization (IM) effect. The IM effect occurs be- +cause, in the early training phase, decreasing the loss values +of inliers rather than outliers is a more beneficial direction to +reduce the overall loss function. Detailed discussions about +the IM effect are given in Section 3.2 and 3.3. Note that +deep generative models does not require class labels, thus +domain-specific techniques such as the annotation method +used in E3-Outlier are not necessary and so our method is +domain-agnostic. +Utilizing the IM effect, we propose a simple but power- +ful OD solver called the outlier detection via the IM effect +(ODIM) to identify outliers from a given training data set. +We train a deep generative model with a log-likelihood- +based approach such as the VAE (Kingma & Welling, 2013) +or IWAE (Burda et al., 2016) for a few updates, and we re- +gard data with large loss values compared to the per-sample +loss distribution as outliers. +As the IM effect is sensitive to how many updates of a deep +generative model proceed, the key to the success of our +method is to choose the optimal number of updates to utilize +the IM effect maximally. For this purpose, we develop the +following strategy. At each update, we fit the Gaussian +mixture model with two components to the per-sample loss +distribution and evaluate the Wasserstein distance of the two +components. We chase the distance as the update proceeds +and select the optimal update point at which the distance +becomes the largest. +Our method has several advantages over existing OD meth- +ods. First, as mentioned above, the ODIM is agnostic to +data domains such as tabular, sequential or image since it is +built on unsupervised learning. By analyzing numerous data +sets, 20 in total, rooted in various domains, we demonstrate +that the ODIM consistently yields competitive or superior +results in identifying outliers. See Section 4. +Second, the ODIM is efficient in computational time and +resources because it requires only a few training updates, +usually a few epochs, in the training phase to detect outliers. +Thus, even when we train multiple generative models to +utilize an ensemble technique, the ODIM is still much faster +than other recent UOD solvers such as (Ruff et al., 2018b; +Lai et al., 2020a) which require at least 200 training epochs. +Third, the ODIM is relatively insensitive to the choice of +the hyper-parameters; thus, it is easy to apply to real prob- +lems without much effort. In contrast, most existing UOD +methods have the objective functions with regularized terms +that should be controlled carefully (Ruff et al., 2018b; 2020; +Lai et al., 2020b). For example, they are even sensitive +to the choice of the learning scheduler. Sensitivity to the +hyper-parameters makes it difficult to use the corresponding +algorithms in practice. +This paper is organized as follows. Section 2 provides a +brief review for related researches for OD problems. The +detailed descriptions of the ODIM algorithm with discus- +sions of the IM effect are given in Section 3. Results of +various experiments including performance tests and abla- +tion studies are presented in Section 4. Further discussions +are provided in Section 5 and concluding remarks follow in +Section 6. +The key contributions of this work are: +• We find a new phenomenon called the IM effect that +deep generative models memorize inliers prior to out- +liers at early training phases. +• We develop a simple and domain-agnostic UOD learn- +ing method called the ODIM to identify outliers in a +given unlabeled training data set contaminated with +anomalous samples. +• We empirically demonstrate the superiority and effi- +ciency of our method by analyzing various benchmark +data sets. +2. Related works +In this section, we only consider SSOD and UOD problems. +Semi-supervised outlier detection +A popular technique +for SSOD is the one class classification approach which +transforms data into a feature space and distinguishes out- +liers from inliers by their radii from the center on the feature +space. The OCSVM (Sch¨olkopf et al., 2001) and SVDD +(Tax & Duin, 2004) are two representative algorithms, which +use kernel techniques to construct the feature space. +Succeeding their ideas, plenty of SSOD algorithms using +deep neural networks have been developed. The DeepSVDD +(Ruff et al., 2018a) extends the SVDD by utilizing a deep + +ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +autoencoder (AE) for learning a feature map, and the Deep- +SAD (Ruff et al., 2020) modifies the DeepSVDD to incor- +porate labeled outliers to training data. Modifications of +the DeepSVDD have been developed by (Zong et al., 2018; +Mahmood et al., 2021; Xia et al., 2015). In addition to AE, +deep generative models are also popularly used for SSOD +(Ryu et al., 2018; Nalisnick et al., 2019; Jiang et al., 2022). +There are methods for SSOD other than the one class clas- +sification approach. The SimCLR (Chen et al., 2020) and +BERT (Devlin et al., 2019) utilize self-supervised learning +by generating artificial labels automatically to obtain a desir- +able feature map, and various algorithms based on this idea +have been developed (Golan & El-Yaniv, 2018a; Bergman +& Hoshen, 2020; Tack et al., 2020; Sehwag et al., 2021). +When some labels (not related to inliers or outliers) are +available, feature maps for classification of those labels can +be used for distinguishing outliers from inliers (Hendrycks +& Gimpel, 2017; Liang et al., 2018; Gomes et al., 2022). +Unsupervised outlier detection +As for traditional ap- +proaches, the LOF (Breunig et al., 2000a) compares the +density of a given datum compared to the densities of its +neighborhoods, and the IF (Liu et al., 2008a) utilizes the fact +that outliers can be separated out by random trees with rela- +tively small sizes. The UOCL (Liu et al., 2014) solves UOD +problems by employing pseudo soft labels and training them +jointly with the one-class classification model. +There are various methods to solve UOD problems with +deep learning models. The RDA (Zhou & Paffenroth, 2017) +combines the robust PCA and AE to detect outliers. The +DSEBM (Zhai et al., 2016) utilizes the energy-based model +for density estimation and uses the energy score or recon- +struction error to identify outliers. The RSRAE (Lai et al., +2020b) devises a new hidden layer called RSR, inserting +it between encoder and decoder of a deep AE to separate +inliers and outliers effectively. The E3-Outlier (Wang et al., +2019a) trains a deep neural network by self-supervised learn- +ing and identifies outliers based on how fast the loss de- +creases as the training proceeds. +3. Proposed method +3.1. Notations and definitions +For a given input vector x ∈ RD, we denote its anomalous- +ness by yo ∈ {0, 1}, that is, yo = 0 if x is an inlier and +yo = 1 otherwise. Note that only x is observable but yo +is not under the UOD task. Let Utr = {x1, . . . , xn} be +unlabeled training data. Our goal is to detect outlier sam- +ples, i.e. x with yo = 1, from Utr as accurately as possible. +Let p(x|z; θ) and q(z|x; φ) be given encoder and decoder +parameterized by θ and φ, respectively, where z ∈ Rd (gen- +erally assuming d < D) is a latent vector. +For a given p ∈ N, we denote the lp-norm of a vector a by +∥a∥p. For two real-valued functions defined on R+, f(t) +and g(t), f(t) is said to be Θ(g(t)) if there exist positive +constants C1, C2 and T such that C1·g(t) ≤ f(t) ≤ C2·g(t) +holds for all t ≥ T. +3.2. Main motivation: inlier-memorization effect +Before proposing our method, we first explain the main +motivation - the inlier-memorization effect. Suppose that +we are training a deep generative model with a certain learn- +ing framework where the training data contain both inliers +and outliers. For an illustration of the IM effect, we ana- +lyze Cardio data set and train a deep generative model +using the VAE method (Kingma & Welling, 2013). The +encoder and decoder architectures are 2-layered deep neural +networks (DNNs) with d = 5 and 50 hidden nodes for each +hidden layer. We are to closely look at the per-sample loss +distribution in an early training phase. For this purpose, we +train the encoder and decoder by minimizing the VAE loss +function only for one epoch. +The middle panel in Figure 1 shows the empirical distri- +bution of the per-sample loss values of the training data. +We can observe that the loss values of inliers tend to be +smaller than those of outliers, which means that the genera- +tive model is trained in the direction of memorizing inliers +first at the beginning of the training phase, and we call this +phenomenon the inlier-memorization (IM) effect. +Conceptually, the IM effect is not a surprising phenomenon. +Assume that the per-sample loss function is continuous on +the input space. Hence, reducing the loss function on dense +regions of the input space is beneficial to reduce the overall +loss function (e.g. the negative log-likelihood). Since inliers +usually locate on dense regions while outliers locate on +sparse regions, reasonable learning algorithms would focus +more on dense regions in an early stage of the training +phase, which results in the IM effect. Note that the IM +effect is observed only in an early training phase since the +learned model memorizes both of inliers and outliers in a +later training phase. +3.3. Theoretical analysis +We provide a theoretical explanation of the inlier- +memorization effect with a toy example where we train +a linear factor model using the VAE. That is, p(x|z; θ) is the +density function of Wz+b+ϵ, where W ∈ RD×d, b ∈ RD +are the loading matrix and bias vector and ϵ ∼ N(0D, σ2ID) +is a noise vector. For q, we set q(z|x; φ) as the density func- +tion of Ux + v + τ, where U ∈ Rd×D, U ∈ Rd, and +τ ∼ N(0d, η2Id). Here, we set σ and η as fixed values. +Thus, θ and φ are (W, b) and (U, v), respectively. Note that +the objective function of the VAE for a given input vector x +is given as + +ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +Figure 1. (Left) The Euclidean norm distribution of Cardio data. There is no significant distributional discrepancy between inliers +and outliers. (Middle) The distribution of the per-sample loss values of Cardio data after training a single epoch. A significant gap +between the distributions of inliers and outliers is seen. (Right) The positive relationship between the Wasserstein distance and identifying +performance (AUC) on Cardio data. +LVAE(θ, φ; x) := Ez∼q(z|x;φ) +� +log +�p(x|z; θ)p(z) +q(z|x; φ) +�� +, +(1) +where p(z) is the density function of the standard multi- +variate Gaussian distribution. We assume that each element +in W, b, U, and v is randomly initialized by the i.i.d. uni- +form distribution on [−1, 1]. Then we have the following +proposition whose proof is given in Appendix. +Proposition 3.1. 1 For a given input vector x, the following +holds: +Eθ,φ +���� +∂ +∂θLVAE(θ, φ; x) +���� +2 +2 += Θ +� +∥x∥4 +2 +� +. +Proposition 3.1 implies that at the beginning of learning, +the Euclidean norm of the gradient of the VAE depends +on the euclidean norm of the input vector. It implies that +if the norms of inliers and outliers are similar, the initial +update direction of θ is influenced more by inliers than +outliers due to their imbalanced proportions. Therefore, in +the early training phase, the generative model is trained in +the direction of memorizing inliers before outliers and thus +the IM effect emerges. +Even though Proposition 3.1 considers a linear generated +model, its implication is still valid for a deep generated +model. To confirm this statement, we conduct a simple +experiment to support our theoretical explanation of the IM +effect by analyzing Cardio data set, already explained in +1Since the generative model p(x; θ) is related only with the +parameter θ, we only consider the gradient with respect to θ. +Section 3.2. We normalize the input features of Cardio +data so that each feature is ranged between 0 and 1. +Using these normalized data, we train a deep generative +model by minimizing the VAE objective function for a single +epoch, and take a look at the per-sample loss distributions of +inliers and outliers. The result is given in the middle panel +of Figure 1 where the per-sample losses of the inliers are +much smaller than those for the outliers. This IM effect can +be explained as follows. The similarity of the distributions +of the norms of inliers and outliers is shown in the left panel +of Figure 1 and Proposition 3.1 implies that the norm of the +per-sample gradient is similar for inliers and outliers and +thus the initial update direction of θ is determined mainly +by the inliers (because the number of inliers is much larger) +to yield the IM effect. +3.4. Outlier detection via inlier-memorization effect: +Algorithm +The IM effect provides a way of detecting outlier by utilizing +the per-sample loss values of a deep generative model at +an early training phase. In this subsection, we propose a +new learning algorithm for solving UOD problems, which +is called the outlier detection via the inlier-memorization +effect (ODIM). The ODIM algorithm consists of two steps: +1) train a deep generative model for a pre-specified number +of updates and 2) based on the per-sample loss values, regard +a given sample as an outlier when the corresponding loss +value is large. To implement this idea, a couple of additional +techniques are needed which are explained below. +Choice of the learning algorithm for a deep generative +model +We should choose a learning algorithm for a deep + +2.5 - +Normal +Anomaly +2.0 +1.5 +1.0 +0.5 +0.0 +1.0 +1'5 +2.0 +2.5Normal +Anomaly +0.8 +0.6 +0.4 +0.2 +0.0 +10 +1112 +13 +140.95 +0.90 +0.85 +0.80 +0.75 +0.70 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +Wass. Dist.ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +Figure 2. Comparison results of (Left) AUC and (Right) AP values varying the pollution rate r from 0.1 to 0.3. We analyze two data sets: +(Upper) MNIST and (Lower) FMNIST. Vertical bars present the standard deviations. +Figure 3. Distributions of per-sample (Left) data norms and (Right) gradient norms on Cardio. We consider two pre-processing schemes +to normalize each feature: 1) (Upper) min-max pre-processing, and 2) (Lower) standardization pre-processing. +generative model carefully to make the IM effect appear +more clearly. There exist numerous algorithms to train a +deep generative model, which can be roughly divided into +two approaches: 1) maximizing the log-likelihood (Kingma +& Welling, 2013; Burda et al., 2016; Rezende & Mohamed, +2015; Tomczak & Welling, 2018; Kim et al., 2020) and 2) us- +ing adversarial networks (Goodfellow et al., 2014; Nowozin +et al., 2016; Arjovsky et al., 2017; Gulrajani et al., 2017). +It is well-known that adversarial-network-based methods +provide more realistic data. But due to their inherent feature +of searching the saddle points (i.e. solving the min-max +problem), it is difficult to define the per-sample loss and +thus it is not suitable for the ODIM. +In contrast, the per-sample loss in the likelihood approach is +naturally defined as the per-sample negative log-likelihood +value, and thus it is easy to develop the ODIM algorithm +with the log-likelihood-based approach. Calculation of the +likelihood function, however, is computationally difficult. +To resolve this problem, we use a computable lower bound +of the log-likelihood, such as the evidence lower bound +(ELBO) used in the variational autoencoder (VAE, (Kingma +& Welling, 2013)). +There exist several lower bounds on the log-likelihood +tighter than ELBO (Burda et al., 2016; Tomczak & Welling, +2018; Kim et al., 2020). Among them, we decide to employ +the importance weighted autoencoder (IWAE, (Burda et al., +2016)) because IWAE can control the tightness of its lower +bound to the log-likelihood and is relatively easy to com- +pute. Usually, the IM effect becomes more obvious when +the lower bound is tighter, but more computation is required +to make the lower bound be tighter. + +0.5 +Normal +Anomaly +0.4 +0.3 +0.2 +0.1 +0.0 +5 +10 +200.12 +Normal +、 +Anomaly +0.10 +0.08 +0.06 +0.04 +0.02 +0.00 +0 +50 +100 +150 +200 +2500.90 +Ours +IF +0.85 +OCSVM +LoF +0.80 +RSRAE +DeepSVDD +0.75 +0.70 +0.65 +0.60 +0.55 +0.1 +0.15 +0.2 +0.25 +0.30.7 +Ours +IF +OCSVM +0.6 +LoF +RSRAE +DeepSVDD +0.5 +0.4 +0.3 +0.1 +0.15 +0.2 +0.25 +0.3Ours +0.9 +IF +OCSVM +LoF +0.8 +RSRAE +DeepSVDD +0.7 +0.6 +0.5 +0.1 +0.15 +0.2 +0.25 +0.30.8 +Ours +0.7 +IF +OCSVM +0.6 +LoF +RSRAE +0.5 +DeepSVDD +0.4 +0.3 +0.2 +0.1 +0.1 +0.15 +0.2 +0.25 +0.32.5 - +Normal +Anomaly +2.0 +1.5 +1.0 +0.5 +0.0 +1.0 +1'5 +2.0 +2.50.35 +Normal +OE0 +Anomaly +0.25 +0.20 +0.15 +0.10 +0.05 +0.00 +10 +15 +20 +25ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +Figure 4. (Left) AUC results on tabular data sets with various number of samples drawn from q(z|x; φ). (Middle) AUC results on tabular +data sets with various number of models from one to twenty to take ensemble. (Right) AUC results on FMNIST for each class with +various learning rates. We vary the learning rate from 1e-4 to 1e-1. +Table 1. Description of tabular data sets +Data +Size +# features +# outliers (%) +Data +Size +# features +# outliers (%) +BreastW +683 +9 +239 (35%) +Vowels +1456 +12 +50 (3.4%) +Cover +286048 +10 +2747 (0.9%) +Wbc +278 +30 +21 (5.6%) +Glass +214 +9 +9 (4.2%) +Arrhythmia +452 +274 +66 (15%) +Ionosphere +351 +33 +126 (36%) +Cardio +1831 +21 +176 (9.6%) +Mammography +11183 +6 +260 (2.32%) +Satellite +6435 +36 +2036 (32%) +Musk +3062 +166 +97 (3.2%) +Satimage-2 +5803 +36 +71 (1.2%) +Pendigits +6870 +16 +156 (2.27%) +Shuttle +49097 +9 +3511 (7%) +Pima +768 +8 +268 (35%) +Thyroid +3772 +6 +93 (2.5%) +The objective function of the IWAE is given as: +LIWAE(θ, φ; x) +:=Ez1,...,zK∼q(z|x;φ) +� +log +� +1 +K +K +� +k=1 +p(x|zk; θ)p(zk) +q(zk|x; φ) +�� +, +(2) +where p(z) is the density of the standard multivariate Gaus- +sian distribution and K is the number of samples. Note that +the IWAE reduces to the VAE when K = 1. In practice, the +IWAE utilizes the Monte Carlo method to approximate (2) +by +�LIWAE(θ, φ; x) := log +� +1 +K +K +� +k=1 +p(x, z; θ)p(zk) +q(zk|x; φ) +� +, +(3) +where zk, k = 1, . . . , K are independently drawn from +q(z|x; φ). We train the encoder and decoder simultaneously +by maximizing the empirical expectation of (3) +Ex∼Utr +� +�LIWAE(θ, φ; x) +� += 1 +n +n +� +i=1 +�LIWAE(θ, φ; xi), +(4) +with respect to θ and φ simultaneously over Utr. +It is known that LIWAE(θ, φ; x) converges to the log- +likelihood as K goes to infinity (Burda et al., 2016). How- +ever, a larger K requires more computation and hence we +need to choose K carefully to compromise performance and +computation. In this study, we set the value of K to 50. An +ablation study for the role of K is done whose results are +given in Section 4.4. +Choice of the optimal number of updates +We empiri- +cally find out that the IM effect usually appears at very early +in the training phase, even sometimes fewer than a single +epoch, and the degree of the IM effect (i.e., the difference of +the loss distributions between inliers and outliers) depends +sensitively on the number of updates of the model. More- +over, the optimal number of updates varies from data to +data. Thus, it would be a key for the success of the ODIM +algorithm to choose the optimal number of updates data +adaptively. +We devise a heuristic strategy to decide the number of up- +dates where the IM effect is maximized. At each update of +the model, we assess the degree of bimodality of the per- +sample loss distribution and select the optimal number of +updates at which the degree of bimodality is maximized. +To be more specific, let l1, . . . , ln be n be the normalized +loss values calculated with the current estimated generative +model, having values between 0 and 1, of the training data. +We fit the two component Gaussian mixture model (GMM- +2) π1N(µ1, σ2 +1) + π2N(µ2, σ2 +2) on these loss values, and +then we investigate how much the two normal distributions +in the fitted Gaussian mixture model are different to measure +the degree of bimodality. For the discrepancy measure of +two normal distributions, we use the Wasserstein distance + +Glass +Mammography +0.9 +Pendigits +Pima +Vowels +0.8 +Wbc +Cardio +Thyroid +0.7 +0.6 +1 +2 +5 +10 +20 +50 +70 +1000.96 +Cover +0.94 +lonosphere +Pendigits +0.92 +Vowels +Wbc +0.90 +Cardio +Thyroid +0.88 +0.86 +0.84 +0.82 +1 +2 +5 +10 +12 +15 +201.0 +0 +0.9 +0.8 +4 +n +0.7 +0.6 +8 +0.5 +0.4 +0.3 +1e-4 2.5e-4 5e-4 +1e-3 2.5e-3 5e-3 +1e-2 2.5e-2 5e-2 +1e-1ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +Algorithm 1 ODIM +In practice, we set (K, Nu, Npat) = (50, 10, 10). +Input: Training data set Utr = {x1, ..., xn} +Require: : Decoder: p(x|z; η), Encoder: q(z|x; φ), GMM-2: π1N(µ1, σ2 +1) + π2N(µ2, σ2 +2), Mini-batch size: nmb, Opti- +mizer: O, Number of samples in IWAE: K, Update unit number: Nu, Maximum patience: Npat +Require: : Auto-encoder: p(x|z; η) and q(z|x; φ), Classifier: f(x; θ), GMM-2: π1N(µ1, σ2 +1) + π2N(µ2, σ2 +2) +while npat < Npat do +for k in 1 : Nu do +1. Generate a subset U with the mini-batch size of nmb from Utr. +2. Using the mini-batch U and optimizer O, update θ and φ with the mini-batch version of (4). +li ← �LIWAE(xi; θ, φ) for i = 1, . . . , n. +// per-sample loss +{˜li}n +i=1 ← normalize({li}n +i=1). +// normalize loss values +Fit the parameters in GMM-2 using {˜li}n +i=1. +if dWD > Dmax +WD then +Dmax +WD ← dWD +// update the maximum Wasserstein distance +l∗ +i ← li, i = 1, . . . , n +// update the best inlier scores +θ∗, φ∗ ← θ, φ +// save the best parameters +npat ← 0 +else +npat ← npat + 1 +end if +end for +end while +Output: l∗ +i , i = 1, . . . , n +// best inlier scores +Output: θ∗, φ∗ +// best parameters +which is given as +W2(N(µ1, σ2 +1), N(µ2, σ2 +2)) = (µ1 − µ2)2 + (σ1 − σ2)2. +(5) +The right panel in Figure 1 illustrates the values of AUC +on the training data of Cardio at the first 10 × m updates +for m = 1, . . . , 50 and their corresponding Wasserstein +distances. We can clearly see that the Wasserstein distance +is a useful measure for selecting the optimal number of +updates. +In practice, we calculate the Wasserstein distance at ev- +ery Nu update and terminate the update when the largest +Wasserstein distance has not been improved for Npat many +calculations of the Wasserstein distance in a row, and se- +lect the optimal number of updates as the one at which the +Wasserstein distance is maximized. We set Nu and Npat to +10 in all the following numerical experiments unless specifi- +cally stated. +We summarize the ODIM’s pseudo algorithm in Algorithm +1. +Ensemble of ODIM scores +To improve and stabilize our +method, we adopt an ensemble strategy. We train multiple +models to have multiple best models for detecting outliers. +From the best models, we obtain multiple per-sample loss +values for each datum, and take the average of them to yield +the final score. The number of multiple models is fixed +to 10 in our experiments. We will empirically show the +effectiveness of using multiple models in Section 4.4. +Since the IM effect is maximized at early in the training +phase and learning multiple models can be done in parallel, +the ensemble ODIM is still faster than other deep learning +algorithms which require hundreds of training epochs. +4. Numerical experiments +In this section, we show the superiority of our proposed +method empirically through extensive experiments. We an- +alyze numerous data sets, 20 in total, that cover tabular, +image, and sequential types. We show that the performance +of the ODIM for detecting outliers is favorably compared +with other competitors regardless of data types. We also +carry out ablation studies including the running time analy- +sis and effect of hyper-parameters. +In the experiments, we report the averaged results based +on five trainings with randomly initialized parameters. We +apply the Pytorch framework to implement our algorithm +using a single NVIDIA TITAN XP GPU. We leave our +implementation code on our GitHub web page. 2 +2https://github.com/jshwang0311/ODIM + +ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +Table 2. Training AUC value comparisons on tabular data sets +Data +IF +OCSVM +LOF +DeepSVDD +RSRAE +Ours +BreastW +0.983 +0.817 +0.447 +0.863 +0.507 +0.992 +Cover +0.894 +0.914 +0.553 +0.522 +0.820 +0.837 +Glass +0.714 +0.265 +0.802 +0.815 +0.480 +0.725 +Ionosphere +0.859 +0.758 +0.894 +0.827 +0.969 +0.914 +Mammography +0.862 +0.846 +0.754 +0.696 +0.535 +0.848 +Musk +0.999 +0.816 +0.370 +0.759 +0.726 +1.000 +Pendigits +0.957 +0.936 +0.486 +0.613 +0.907 +0.953 +Pima +0.672 +0.626 +0.648 +0.027 +0.587 +0.706 +Vowels +0.807 +0.607 +0.946 +0.787 +0.876 +0.904 +Wbc +0.938 +0.934 +0.937 +0.763 +0.598 +0.941 +Arrhythmia +0.810 +0.807 +0.786 +0.677 +0.814 +0.800 +Cardio +0.931 +0.913 +0.716 +0.589 +0.245 +0.916 +Satellite +0.690 +0.598 +0.558 +0.637 +0.731 +0.690 +Satimage-2 +0.992 +0.979 +0.491 +0.739 +0.988 +0.997 +Shuttle +0.997 +0.983 +0.508 +0.646 +0.972 +0.981 +Thyroid +0.977 +0.847 +0.942 +0.781 +0.815 +0.928 +4.1. Data set description +Tabular data sets +We consider 16 tabular data sets which +are frequently analyzed in the OD literature. They are ob- +tained from various domains, such as clinical pathology or +astronomy, and are publicly accessible on (Rayana, 2016). +We refer Table 1 for the detailed descriptions of the data +sets. All the data sets have the labels about the anomalous +information, but we exclude them in the training phase to +conduct unsupervised learning tasks, but use them in the +test phase to assess the accuracy of outlier detection. +MNIST & Fashion MNIST +We analyze two synthetic +image data sets made from MNIST (LeCun et al., 1998) +and FMNIST (Xiao et al., 2017). MNIST and FMNIST +contain grey-colored 28 × 28 images of handwritten digits +and clothing, respectively. Both data sets consist of 60K +training data and 10K test data. +To conduct UOD tasks, we pre-process the two data sets in +advance, as is done in the previous works (Ruff et al., 2018b; +Chalapathy et al., 2018; Golan & El-Yaniv, 2018b; Wang +et al., 2019b). We choose a class c to be the normal class +and regard the others as abnormal classes. We select all the +training images whose class is c and randomly draw samples +from the remaining data so that the ratio between the number +of inlier and outlier data is 1 : r, where r ∈ [0, 1] is a +pre-specified pollution rate. Then, we combine the normal +and abnormal data and discard their class labels to make +unlabeled data. Note that both MNIST and FMNIST have +ten classes; thus, we can generate ten types of training data +with the pollution rate r. We report averaged performance +on the ten training data unless otherwise stated. +WM-811K +We also investigate the real-world image data +set, WM-811K (Wu et al., 2014), containing wafer im- +ages obtained from the semiconductor fabrication process. +Among 811K images, we pick a part of images with label in- +formation about whether a given image is a failure (outlier). +As a result, we utilize 172,950 wafers that include 25,519 +failure images. Since the data have diverse sizes, we resize +them to have the unified shape of 28 × 28 before analyzing +them. +Reuters-21578 +Besides tabular and image data, we addi- +tionally analyze text type data. Reuters-21578 data set +is a collection of 21,578 documents from Reuters newswire +in 1987. We follow the procedure in (Lai et al., 2020a) +to make analyzable data. We use the pre-processed data +with a fixed dimension of 26,147 by applying the TFIDF +transformer. We consider the five largest classes among +90 classes and randomly draw 360 samples randomly from +each class. Similar to MNIST and FMNIST, we build train- +ing data using a pre-specified pollution rate r for each class +and report averaged results over the five classes. +4.2. Implementation details +Data pre-process +In addition to identifying outliers in +a given training data set, we also assess the identification +performance in unseen (test) data. We use the given test +data for MNIST and FMNIST. For the other data sets, we +randomly split each data set into two partitions with the rate +of 6:4 and use the first and second partitions as training and +test data sets, respectively. +After the training and test data are constructed, we apply +the min-max normalization to them so that each feature has + +ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +Table 3. Training AP value comparisons on tabular data sets +Data +IF +OCSVM +LOF +DeepSVDD +RSRAE +Ours +BreastW +0.961 +0.843 +0.308 +0.678 +0.392 +0.988 +Cover +0.049 +0.085 +0.012 +0.012 +0.149 +0.032 +Glass +0.080 +0.032 +0.170 +0.149 +0.104 +0.119 +Ionosphere +0.821 +0.730 +0.850 +0.757 +0.951 +0.867 +Mammography +0.211 +0.108 +0.103 +0.053 +0.024 +0.098 +Musk +0.999 +0.124 +0.025 +0.132 +0.057 +1.000 +Pendigits +0.291 +0.211 +0.038 +0.039 +0.215 +0.302 +Pima +0.507 +0.456 +0.446 +0.431 +0.427 +0.491 +Vowels +0.149 +0.085 +0.442 +0.124 +0.227 +0.375 +Wbc +0.560 +0.553 +0.557 +0.206 +0.113 +0.710 +Arrhythmia +0.448 +0.415 +0.394 +0.362 +0.518 +0.443 +Cardio +0.536 +0.543 +0.195 +0.140 +0.059 +0.564 +Satellite +0.673 +0.583 +0.389 +0.461 +0.688 +0.652 +Satimage-2 +0.902 +0.847 +0.027 +0.031 +0.887 +0.949 +Shuttle +0.980 +0.951 +0.084 +0.167 +0.713 +0.947 +Thyroid +0.541 +0.148 +0.239 +0.124 +0.186 +0.327 +Table 4. Averaged ranks of AUC and AP over tabular data sets +IF +OCSVM +LOF +DeepSVDD +RSRAE +Ours +AUC +2.094 +3.813 +4.250 +4.750 +4.000 +2.094 +AP +2.313 +3.813 +4.250 +4.813 +3.688 +2.125 +the minimum zero and maximum one. We mainly focus on +the detecting performance on training data, and present the +performance on test data in Section 4.4 of ablation study. +Architecture & learning schedule +We use two hidden +layered DNN architectures for building the encoder and +decoder and set K, the number of samples drawn from the +encoder used for constructing the IWAE objective function, +to 50. We optimize the IWAE loss function with the Adam +optimizer (Kingma & Ba, 2014) with the mini-batch size +of 128 and the learning rate of 5e-4. To run the ODIM, we +fix the two hyper-parameters, Nu and Npat, to 10. For the +ensemble learning, we train 10 pairs of encoder and decoder, +each of which is trained from different initialized param- +eter values. See our GitHub web page for the detailed +implementations. +Baseline +For baselines to be compared with the ODIM, +we consider three machine-learning-based methods: 1) iso- +lation forests (IF, (Liu et al., 2008b)), 2) one-class SVM +(OCSVM, (Sch¨olkopf et al., 2001)), and 3) local outlier fac- +tor (LOF, (Breunig et al., 2000b)), and two deep-learning- +based methods: 4) deep support vector data description +(DeepSVDD, (Ruff et al., 2018b)) and 5) robust subspace +recovery autoencoder (RSRAE, (Lai et al., 2020a)). +As for the LOF, OCSVM, and IF methods, we implement +them using the scikit-learn package, and we refer to the +official GitHub websites of the DeepSVDD 3 and RSRAE +4 for their re-implementations. Like the ODIM, we report +their average results on five runs with different initials. +4.3. Performance for outlier identification +We first compare the ODIM with the other baselines in +terms of the performance for identifying outliers in a given +training data set. We evaluate the two performance scores: +the area under receiver operating characteristic (AUC or +ROAUC) and average precision (AP). +Results for tabular data +Tables 2 and 3 list the AUC and +AP results on the tabular data sets (We mark the best score +for each data set as the bold face.). The results amply show +the superiority of the ODIM compared to other baselines +in identifying outliers since the ODIM achieves the best +scores most frequently on both AUC and AP (6 for each). +For example, on BreastW and Musk, the ODIM separates +inliers and outliers (almost) perfectly. +In addition, it is notable that the AP values of the ODIM +are not much different from the best scores when it is not +the best, while the AP values of the others have large vari- +ations. In particular, the two deep learning based methods, +DeepSVDD and RSRAE, show relatively unstable results. +For example, the AP values of RSRAE for BreastW and +Cardio are much smaller than those of the ODIM. +To check further the stability of the performance for detect- +ing outliers according to the data sets, we rank the AUC and +AP scores of the algorithms for each data set and take the av- +3https://github.com/lukasruff/Deep-SVDD-PyTorch +4https://github.com/dmzou/RSRAE + +ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +Table 5. Results of AUC and AP over the WM-811K data set. We write the standard deviations of each method inside the parentheses. +IF +OCSVM +LOF +DeepSVDD +RSRAE +Ours +AUC +0.680 (0.004) +0.731 +0.327 +0.505 (0.114) +0.672 (0.011) +0.747 (0.005) +AP +0.354 (0.008) +0.418 +0.072 +0.151 (0.063) +0.203 (0.019) +0.379 (0.003) +Table 6. Results of AUC and AP over the Reuters-21578 data set. We write the standard deviations of each method inside the +parentheses. +IF +OCSVM +LOF +DeepSVDD +RSRAE +Ours +AUC +0.574 (0.063) +0.878 +0.812 +0.501 (0.033) +0.915 (0.022) +0.888 (0.043) +AP +0.202 (0.051) +0.536 +0.388 +0.130 (0.022) +0.610 (0.063) +0.553 (0.096) +erage of the ranks for each algorithm to obtain the averaged +rank, which is summarized in Table 4. The ODIM achieves +the best averaged ranks on both AUC and AP. These stability +results imply that the ODIM can be used as an off-the-shelf +method for outlier identification of tabular data. In the sub- +sequent analyses, we demonstrate that the ODIM is still +stable when dealing with other types of data sets. +MNIST & FMNIST results +Figures 2 compare the re- +sults for MNIST and FMNIST data sets. Similar to the +tabular data, the ODIM works quite well for image data. +Our method achieves the second best performances for all +considering pollution rates on MNIST and the best perfor- +mances on FMNIST. +WM-811K results +We analyze an image data set called +WM-811K consisting of various wafer images. Table 5 +shows that our method marks the best and the second-best +results on AUC and AP, respectively. In contrast, the two +deep-learning-based methods do not work well. +Reuters-21578 results +We analyze Reuters to check +the efficiency of our method on super-high-dimensional +data analysis, whose results are summarized in Table 6. The +ODIM records the second-best AUC and AP values suc- +ceeding the RSRAE. Meanwhile, the IF, which is compared +favorably to the ODIM for tabular data, shows sub-optimal +performances. +Conclusion +Throughout the empirical experiments, we +have noticed that among the various methods for outlier +identification, the ODIM is the only one that performs con- +sistently well regardless of data types. +4.4. Ablation study +Number of samples used in the IWAE +Recall that the +IWAE objective function uses multiple samples, z1, . . . , zK, +independently drawn from q(z|x; φ), to achieve a tighter +lower bound of the log-likelihood. We empirically check +that using a tighter bound actually yields a clearer IM effect. +The left panel in Figure 4 summarizes the AUC values on +several tabular data sets with various Ks from 1 to 100. +For the results of the other tabular data sets, see Appendix. +Note that the IWAE with K = 1 equals the original VAE. +As expected, a lower bound closer to the log-likelihood +tends to provide more obvious IM effect, leading to better +identification performances. We can also observe that when +the value of K becomes larger than 50, the enhancement +seems saturated. For this reason, we set K = 50 in our +experiments. +Implementation time +We investigate how fast our +method is executed compared to other competitors. Table 7 +summarizes the running time comparisons on several data +sets, including tabular and image data. As for our method, +we only display the running time of a single model learning +since training multiple models to make an ensemble can be +done in parallel. The results for the other data sets are listed +in Appendix. +As expected, the two deep learning methods are much slower +than the non-deep-learning-based anomaly detection meth- +ods. Our method is generally faster than the deep learning +methods. In particular, our method is 45 and 82 times faster +than the RSRAE on FMNIST, and Wafer, respectively. +This is surprising because our method identifies outliers +better than the RSRAE on these data sets. When analyzing +data with large sample sizes such as Cover, FMNIST, and +Wafer, our method is similar or even faster than non-deep- +learning methods. +Our method has relatively inferior performance in running +time on Reuters. Unlike the other data sets, Reuters +is a super high-dimensional data set with more than 26,000 +features. We conjecture that the high dimensionality of data +might put off the occurrence of the IM effect. Accelerating +the IM effect to overcome the running time issue when ana- +lyzing super high-dimensional data should be done, which +we will do in a near future. +Number of models to ensemble +We investigate the effect +of the ensemble in the ODIM. We vary the number of models + +ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +Table 7. Running time comparison for the ODIM and other competitors. All records are measured in seconds. +Data +IF +OCSVM +LOF +DeepSVDD +RSRAE +Ours +Cover +6.192 +2164.270 +24.959 +3463.820 +2451.950 +358.378 +Mammography +0.408 +2.482 +0.221 +135.958 +98.356 +21.002 +Pendigits +0.350 +1.246 +1.899 +83.944 +62.739 +15.998 +Satellite +0.369 +1.152 +1.735 +78.876 +71.493 +13.846 +Satimage-2 +0.361 +0.916 +1.340 +72.048 +69.147 +8.580 +Shuttle +0.987 +63.146 +4.426 +594.588 +427.291 +61.471 +FMNIST +4.298 +52.114 +15.446 +744.652 +1495.926 +32.859 +Wafer +89.499 +11498.385 +910.600 +4561.028 +12363.401 +149.986 +Reuters +4.984 +106.567 +1.394 +16.567 +23.931 +92.486 +Table 8. Test AUC results of the ODIM. For MNIST, FMNIST, +and Reuters, we set r = 0.1. +Data +IF +OCSVM +LOF +Ours +BreastW +0.986 +0.880 +0.458 +0.994 +Cover +0.876 +0.915 +0.554 +0.852 +Glass +0.563 +0.680 +0.720 +0.725 +Ionosphere +0.850 +0.736 +0.890 +0.861 +Mammography +0.613 +0.485 +0.848 +0.861 +Musk +0.869 +0.840 +0.720 +1.000 +Pendigits +0.999 +0.821 +0.318 +0.946 +Pima +0.650 +0.527 +0.557 +0.714 +Vowels +0.960 +0.948 +0.418 +0.870 +Wbc +0.691 +0.651 +0.638 +0.936 +Arrhythmia +0.493 +0.515 +0.547 +0.817 +Cardio +0.348 +0.435 +0.482 +0.919 +Satellite +0.747 +0.458 +0.957 +0.692 +Satimage-2 +0.924 +0.900 +0.914 +0.995 +Shuttle +0.824 +0.756 +0.664 +0.985 +Thyroid +0.926 +0.922 +0.711 +0.921 +MNIST +0.825 +0.857 +0.693 +0.843 +FMNIST +0.899 +0.873 +0.496 +0.905 +Wafer +0.683 +0.732 +0.329 +0.723 +Reuters +0.484 +0.918 +0.829 +0.859 +used in the ensemble from one to twenty and compare the +performances on several tabular data sets, whose results are +presented in the middle panel of Figure 4. The results for +the other data sets are summarized in Appendix. +There is a general tendency that using more models helps +improve the identifying performance. The optimal number +of models varies according to data sets, but the performance +is not sensitive to the number of models used in the ensemble +unless it is too small. +Learning schedule +We evaluate the robustness of the +ODIM to the learning schedule. We consider the Adam +optimizer with various learning rates from 1e-4 to 1e-1, +whose results on FMNIST are depicted in the right panel of +Figure 4. We present the results of the ten classes separately, +Table 9. Test AP results of the ODIM. For MNIST, FMNIST, and +Reuters, we set r = 0.1. +Data +IF +OCSVM +LOF +Ours +BreastW +0.968 +0.890 +0.311 +0.988 +Cover +0.056 +0.095 +0.013 +0.034 +Glass +0.140 +0.122 +0.213 +0.152 +Ionosphere +0.803 +0.729 +0.857 +0.815 +Mammography +0.261 +0.109 +0.066 +0.106 +Musk +0.997 +0.126 +0.024 +1.000 +Pendigits +0.300 +0.241 +0.028 +0.295 +Pima +0.537 +0.517 +0.478 +0.539 +Vowels +0.120 +0.058 +0.517 +0.266 +Wbc +0.605 +0.332 +0.353 +0.549 +Arrhythmia +0.490 +0.324 +0.281 +0.403 +Cardio +0.583 +0.525 +0.207 +0.612 +Satellite +0.679 +0.593 +0.376 +0.659 +Satimage-2 +0.925 +0.872 +0.029 +0.959 +Shuttle +0.951 +0.950 +0.061 +0.837 +Thyroid +0.587 +0.065 +0.068 +0.392 +MNIST +0.974 +0.978 +0.954 +0.976 +FMNIST +0.984 +0.982 +0.905 +0.987 +Wafer +0.423 +0.496 +0.109 +0.451 +Reuters +0.782 +0.971 +0.933 +0.923 +where each class is regarded as the inlier class. Note that +the identifying performances rarely change until we use a +learning rate larger than 1e-2. As we usually do not consider +a learning rate much larger than 1e-3 when we apply the +Adam optimizer, we can conclude that our method is stable +with respect to the learning schedule, which implies that our +method can be used in practice without delicate settings. +Identifying unseen data +As we mentioned in Section 4.2, +we have left some portion of data to examine outlier identifi- +cation ability for unseen data. Table 8 and 9 summarize the +test AUC and AP results. On tabular data sets, the results for +unseen data are similar to those for training data, indicating +that our method identifies normal samples well from unseen +data. + +ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +Table 10. Comparison of two data pre-processing methods: 1) min- +max and 2) standardization. We report the AUC results. +Data +Min-max +Standardization +BreastW +0.992 +0.664 +Cover +0.837 +0.961 +Glass +0.725 +0.630 +Ionosphere +0.914 +0.863 +Mammography +0.848 +0.767 +Musk +1.000 +0.799 +Pendigits +0.953 +0.941 +Pima +0.706 +0.437 +Vowels +0.904 +0.639 +Wbc +0.941 +0.856 +Arrhythmia +0.800 +0.769 +Cardio +0.916 +0.943 +Satellite +0.690 +0.740 +Satimage-2 +0.997 +0.969 +Shuttle +0.981 +0.995 +Thyroid +0.928 +0.965 +Table 11. Training AUC values with various values of l. We con- +sider three values for l, l = 0.0, 0.3, 0.5. +Data +l = 0.0 +l = 0.3 +l = 0.5 +Arrhythmia +0.800 +0.837 +0.888 +Cardio +0.916 +0.991 +0.993 +Satellite +0.690 +0.868 +0.881 +Satimage-2 +0.997 +0.998 +0.999 +Shuttle +0.981 +0.990 +0.990 +Thyroid +0.928 +0.995 +0.995 +Interestingly, the test AP results for the other complex data +sets, image and sequential data sets, tend to be higher than +those for training data sets regardless of learning methods. +We think that for such data the distributions of outliers for +training and test data sets are quite different compared to +the those of inliers due to their high dimensionality, which +results in larger loss values for test outliers to make detecting +them easier. +5. Further discussion +5.1. Effect of data pre-processing on the ODIM +Proposition 3.1 indicates that the gradient norm is propor- +tional to the input norm. An interesting fact is that the input +norm is not invariant to the normalization process, and so is +the IM effect. This seemingly counter-intuitive phenomenon +can be explained as follows. Since the initial parameters +in the model considered in Proposition 3.1 are independent +mean 0 random variables, data far from the origin are not +explained well by an initial linear factor model, and thus, +the norms of the corresponding gradients become larger. +Table 12. Training AP values with various values of l. We consider +three values for l, l = 0.0, 0.3, 0.5. +Data +l = 0.0 +l = 0.3 +l = 0.5 +Arrhythmia +0.443 +0.767 +0.772 +Cardio +0.564 +0.934 +0.943 +Satellite +0.652 +0.849 +0.849 +Satimage-2 +0.949 +0.954 +0.958 +Shuttle +0.947 +0.977 +0.979 +Thyroid +0.327 +0.844 +0.845 +The above observation provides an important implication +regarding the IM effect. The IM effect depends on the +choice of the normalization process. For illustration, we +apply the ODIM to the standardization scaling - a normal- +ization process to force every feature to have mean zero +and variance one. Figure 3 compares the input norms and +gradient norms of the initial model of the min-max scaled +data and the standardization scaled data. It is observed that +the input norms of inliers in the min-max scaled data are +similar to those of outliers while the input norms of inliers +in the standardization scaled data are smaller. In turn, as +implied by Proposition 3.1, the gradient norms of inliers +are similar to those of outliers for the min-max scaled data +while the gradient norms of outliers are much larger for the +standardization scaled data. Note that inliers in the standard- +ization scaled data mostly locate around the origin and thus +the input norms of inliers become smaller. +Smaller gradients of inliers generally result in the perfor- +mance degradation. In Table 10, we empirically observe +that among 16 tabular data sets, the ODIM with the stan- +dardization scaled data has better results only on five data +sets. +Even though it is better than the standardization scaling, the +min-max scaling is by no means optimal for the IM effect. +We leave the optimal choice of the normalization process +and/or choice of initial parameters as a future research topic. +5.2. ODIM with labeled data +When outlier labels are available, some studies have ex- +ploited this additional information to solve outlier detection +tasks more efficiently (Ruff et al., 2020; Daniel et al., 2019). +But as far as we know, these existing works require that all +outliers should be labeled, which is equivalent to the SOD +setting. The only difference of (Ruff et al., 2020; Daniel +et al., 2019) compared to the conventional SOD solvers is +that they cast the problem into an one-class classification +problem rather than a two-class one. +While it is very costly to obtain perfectly labeled data, +partially labeled data are frequently met in practice. In +this subsection, we modify the ODIM for such a situa- + +ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +tion. That means, apart from the unlabeled training data set +Utr = {x1, . . . , xn}, we also have a labeled outlier data set +Ltr = {(xl +1, 1), . . . , (xl +m, 1)}. Note that there exist outliers +in Utr. +We simply adopt the idea of (Daniel et al., 2019), which +encourages the log-likelihood of known outliers to decrease +with the variational upper bound. For u > 1, the upper +bound, called χ upper bound (CUBO), is given as: +LCUBO(θ, φ; x) +:= 1 +u log Ez∼q(z|x;φ) +��p(x|z; θ)p(z) +q(z|x; φ) +�u� +. +(6) +With the above CUBO term (6), we modify the loss function +of the ODIM by subtracting the expected CUBO on Ltr +from the original IWAE loss function to have: +Ex∼Utr �LIWAE(θ, φ; x) − γ · Ex∼Ltr �LCUBO(θ, φ; x), +where +�LCUBO(θ, φ; x) := +�p(x|z; θ)p(z) +q(z|x; φ) +�u +, +for z being a random sample drawn from q(z|x; φ) and +γ > 0 is a tuning parameter controlling the degree of the +CUBO loss. The CUBO loss generally increases the IWAE +per-sample loss values of outliers, leading to separating out +the two components of the GMM-2 more clearly. In our +paper, we set the value of γ to one. +To investigate the improvements of our modified method, +we assess the training AUC and AP values over several +tabular data sets with various proportions of labeled outliers, +which are summarized in Table 11 and 12. Here, l ∈ [0, 1] +means the ratio of the labeled and entire outliers. Note that +the case of l = 0.0 is equivalent to when we only use Utr. It +is clearly seen that using label information helps to enhance +identifying performance with a large margin, especially for +the AP, even when the proportion of the labeled data is +small. +Note that the proposed modification of the ODIM for par- +tially labeled data can be improved further. For example, we +can use the labeled information to select the optimal number +of updates. There would be other rooms to improve the +ODIM for partially labeled data, which we leave for future +works. +6. Concluding remarks +This paper proposed a fast, powerful and easy to use UOD +method called the ODIM. The ODIM is inspired by a new +observation called the IM effect that deep generative models +tend to memorize inliers first when they are trained. Com- +bined with the technique to select the optimal number of +training updates and the ensemble method, we found that +the ODIM identified outliers effectively and efficiently; the +ODIM provided consistently superior results regardless of +data types with much faster running times. +As far as the authors know, there are not many theoretical +studies about the behavior of deep neural networks at the +early training phase. It would be valuable to understand +theoretically when and why the IM effect (or the memoriza- +tion effect) emerges. Based on theoretical understandings, +the suboptimal behavior of the ODIM on super high dimen- +sional data could be improved. + +ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +References +Arjovsky, M., Chintala, S., and Bottou, L. Wasserstein gen- +erative adversarial networks. In International conference +on machine learning, pp. 214–223. 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Note that +Eθ,φ +���� +∂ +∂θ LVAE(θ, φ; x) +���� +2 +2 += +� +i +� +j +Eθ,φ +� +∂ +∂wij LVAE(θ, φ; x) +�2 ++ +� +i +Eθ,φ +� ∂ +∂bi LVAE(θ, φ; x) +�2 +. +We are going to characterize the two terms, Eθ,φ +� +∂ +∂wij LVAE(θ, φ; x) +�2 +and Eθ,φ +� +∂ +∂bi LVAE(θ, φ; x) +�2 +, and combine them +to draw the final conclusion. +w.r.t. wij +Since X|(Z = z) ∼ N +� +Wz + b, σ2� +holds, we have +log p(x|z; θ) += +− 1 +2σ2 +D +� +i=1 +(xi − w′ +iz − bi)2 + const += +− 1 +2σ2 +D +� +i=1 +� +�xi − +d +� +j=1 +wijzj − bi +� +� +2 ++ const, +where wi is the i-th row of W and const is a constant not depending on θ. Therefore, we can obtain the following result of +the log-likelihood with respect to wij: +∂ +∂wij +log (p(x|z; θ)) += 1 +σ2 (xi − w′ +iz − bi) · zj += 1 +σ2 +� +�xizj − wijz2 +j − +� +j′̸=j +wij′zjzj′ − bizj +� +� . + +ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +Hence, the first derivative of the VAE objective function with respect to wij becomes: +∂ +∂wij LVAE(θ, φ; x) += +� +z +1 +σ2 +� +�(xi − bi)zj − wijz2 +j − +� +j′̸=j +wij′zjzj′ − bizj +� +� · q(z|x; φ)dz += 1 +σ2 +� +(xi − bi)(u′ +jx + vj) − wij +�� +u′ +jx + vj +�2 + η2� +− +� +u′ +jx + vj +� � +j′̸=j +wij′ � +u′ +j′x + vj′� +� +� += 1 +σ2 +� +(xi − bi)(u′ +jx + vj) +− +� +u′ +jx + vj +� � +j′ +wij′ � +u′ +j′x + vj′� +− wijη2 +� +� , +where uj is the j-th row of U. By squaring the above term, we have +� +∂ +∂wij +LVAE(θ, φ; x) +�2 += 1 +σ4 +� +(xi − bi)2 � +u′ +jx + vj +�2 ++ +� +u′ +jx + vj +�2 � +j′ +w2 +ij′ +� +u′ +j′x + vj′�2 + w4 +ijη4 ++2 +� +u′ +jx + vj +�2 � +j′′>j′ +wij′wij′′(u′ +j′x + vj′)(u′ +j′′x + vj′′) ++2wijη2 � +u′ +jx + vj +� � +j′ +wij′ � +u′ +j′x + vj′� +−2wijη2(xi − bi) +� +u′ +jx + vj +� +−2(xi − bi) +� +u′ +jx + vj +�2 � +j′ +wij′ � +u′ +j′x + vj′� +� +� . +Now, we will calculate the expected value of the above equation with respect to θ and φ. To do this, we are going to take an +expectation for each term in the RHS of the above equation. Note that, for a random variable X ∼ Unif[−1, 1], E [X] = 0, +E +� +X2� += 1/3, and E +� +X4� += 1/5. Thus, we have +Eθ,φ +� +(xi − bi)2 � +u′ +jx + vj +�2� += Eθ,φ +� +(x2 +i − 2xibi + b2 +i ) +� +(u′ +jx)2 + v2 +j + 2vju′ +jx +�� += Eθ,φ +� +(x2 +i + b2 +i ) +�� +i′ +u2 +ji′x2 +i′ + v2 +j +�� += +� +x2 +i + 1 +3 +� +· +�1 +3 ∥x∥2 +2 + 1 +3 +� +, + +ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +Eθ,φ +� +�� +u′ +jx + vj +�2 � +j′ +w2 +ij′ +� +u′ +j′x + vj′�2 +� +� += Eθ,φ +� +� +�� +i′ +uji′xi′ + vj +�2 � +j′ +w2 +ij′ +�� +i′ +uj′i′xi′ + vj′ +�2� +� += 1 +3Eθ,φ +��� +i′ +uji′xi′ + vj +�4 ++ +�� +i′ +uji′xi′ + vj +�2 � +j′̸=j +�� +i′ +uj′i′xi′ + vj′ +�2� +� += 1 +3Eθ,φ +��� +i′ +uji′xi′ +�4 ++ 6 +�� +i′ +uji′xi′ +�2 +v2 +j + v4 +j ++ +�� +i′ +uji′xi′ + vj +�2 � +j′̸=j +�� +i′ +uj′i′xi′ + vj′ +�2� +� += 1 +3Eθ,φ +�� +i′ +u4 +ji′x4 +i′ + 6 +� +i′′>i′ +u2 +ji′uji′′x2 +i′x2 +i′′ ++6 +�� +i′ +u2 +ji′x2 +i′ +� +v2 +j + v4 +j ++ +�� +i′ +u2 +ji′x2 +i′ + v2 +j +� � +j′̸=j +�� +i′ +u2 +j′i′x2 +i′ + v2 +j′ +�� +� += 1 +3 +� +1 +5 +� +i′ +x4 +i + 2 +3 +� +i′′>i′ +x2 +i′x2 +i′′ + 2 +3 +� +i′ +x2 +i′ + 1 +5 ++ +� +1 +3 +� +i′ +x2 +i′ +� � +j′̸=j +� +1 +3 +� +i′ +x2 +i′ +�� +� += 1 +3 +�1 +3∥x∥4 +2 − 2 +15∥x∥4 +4 + 2 +3∥x∥2 +2 + 1 +5 + (d − 1) +�1 +3∥x∥2 +2 + 1 +3 +�� +, +Eθ,φw2 +ijη4 = η4Eθw2 +ij = 1 +3η4, +Eθ,φ +� +�2 +� +u′ +jx + vj +�2 � +j′′ >j′ +wij′wij′′(u′ +j′x + vj′)(u′ +j′′x + vj′′) +� +� += 0, + +ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +Eθ,φ +� +�2wijη2 � +j′ +wij′ � +u′ +jx + vj +� � +u′ +j′x + vj′� +� +� += +2Eθ,φ +� +w2 +ijη2 � +u′ +jx + vj +�2� += +2 +3η2Eφ +� +� +�� +i′ +uji′xi′ + vj +�2� +� += +2 +3η2Eφ +�� +i′ +u2 +ji′x2 +i′ + v2 +j +� += +2 +9η2 � +∥x∥2 +2 + 1 +� +, +Eθ,φ +� +−2wijη2(xi − bi) +� +u′ +jx + vj +�� += 0, +and +Eθ,φ +� +�−2(xi − bi) +� +u′ +jx + vj +�2 � +j′ +wij′ � +u′ +j′x + vj′� +� +� = 0. +By integrating all the above expected values and using the property ∥x∥2 ≥ ∥x∥4, we arrive at the following result: +Eθ,φ +� +∂ +∂wij +LVAE(θ, φ; x) +�2 += Θ +� +∥x∥4 +2 +� +. +w.r.t. bi +We have +∂ +∂bi +log p(x|z; θ) = 1 +σ2 +� +�xi − +d +� +j=1 +wijzj − bi +� +� , +thus, +∂ +∂bi +LVAE(θ, φ; x) += +� +z +1 +σ2 +� +�xi − +d +� +j=1 +wijzj − bi +� +� · q(z|x; φ)dz += 1 +σ2 +� +�(xi − bi) − +� +j +wij(u′ +jx + vj) +� +� . +By squaring the above term, +� ∂ +∂bi +LVAE(θ, φ; x) +�2 += 1 +σ4 +� +�(xi − bi)2 + +� +j +w2 +ij(u′ +jx + vj)2 ++2 +� +j′>j +wijwij′(u′ +jx + vj)(u′ +j′x + vj′) +− 2(xi − bi) +� +j +wij(u′ +jx + vj) +� +� . + +ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +Here, we calculate the expected value of each term in the above RHS. We have +Eθ,φ +� +(xi − bi)2� += x2 +i + 1 +3, +Eθ,φ +� +�� +j +w2 +ij(u′ +jx + vj)2 +� +� += +1 +3 +� +j +Eθ,φ +�� +i′ +u2 +ji′x2 +i′ + v2 +j +� += +d +9∥x∥2 +2 + d +9, +Eθ,φ +� +�2 +� +j′>j +wijwij′(u′ +jx + vj)(u′ +j′x + vj′) +� +� = 0, +and +Eθ,φ +� +�2(xi − bi) +� +j +wij(u′ +jx + vj) +� +� = 0, +Combining the above expectations, we have +Eθ,φ +� ∂ +∂bi +LVAE(θ, φ; x) +�2 += Θ +� +∥x∥2 +2 +� +. +Final conclusion +Combining the above results, we have +Eθ,φ +���� +∂ +∂θLVAE(θ, φ; x) +���� +2 +2 += +� +i +� +j +Eθ,φ +� +∂ +∂wij +LVAE(θ, φ; x) +�2 ++ +� +i +Eθ,φ +� ∂ +∂bi +LVAE(θ, φ; x) +�2 += +� +i +� +j +Θ +� +∥x∥4 +2 +� ++ +� +i +Θ +� +∥x∥2 +2 +� += Θ +� +∥x∥4 +2 +� +. □ + +ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +Table A.1. AUC results of the ODIM with various values of K. +Data +K = 1 +K = 2 +K = 5 +K = 10 +K = 20 +K = 50 +K = 70 +K = 100 +BreastW +0.982 +0.985 +0.993 +0.990 +0.990 +0.991 +0.991 +0.992 +Glass +0.525 +0.676 +0.636 +0.689 +0.702 +0.704 +0.704 +0.727 +Ionosphere +0.868 +0.854 +0.845 +0.860 +0.870 +0.860 +0.860 +0.866 +Mammography +0.565 +0.794 +0.807 +0.835 +0.841 +0.852 +0.852 +0.853 +Musk +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +Pendigits +0.847 +0.930 +0.922 +0.955 +0.959 +0.950 +0.950 +0.950 +Pima +0.626 +0.663 +0.682 +0.706 +0.696 +0.706 +0.706 +0.705 +Vowels +0.528 +0.655 +0.549 +0.583 +0.875 +0.900 +0.900 +0.896 +Wbc +0.866 +0.929 +0.915 +0.919 +0.937 +0.935 +0.935 +0.939 +Arrhythmia +0.794 +0.769 +0.768 +0.786 +0.778 +0.782 +0.782 +0.786 +Cardio +0.794 +0.919 +0.913 +0.932 +0.925 +0.914 +0.914 +0.919 +Satellite +0.730 +0.756 +0.737 +0.713 +0.707 +0.690 +0.690 +0.688 +Satimage-2 +0.934 +0.995 +0.988 +0.991 +0.994 +0.994 +0.994 +0.995 +Shuttle +0.682 +0.977 +0.972 +0.975 +0.975 +0.978 +0.978 +0.984 +Thyroid +0.636 +0.860 +0.868 +0.887 +0.908 +0.924 +0.924 +0.926 +Table A.2. Running time comparison. +Data +IF +OCSVM +LOF +DeepSVDD +RSRAE +Ours +BreastW +0.198 +0.016 +0.013 +10.528 +7.773 +5.095 +Cover +6.192 +2164.270 +24.959 +3463.821 +2451.950 +358.378 +Glass +0.178 +0.007 +0.008 +2.734 +2.642 +7.072 +Ionosphere +0.256 +0.009 +0.022 +5.023 +5.217 +6.240 +Mammography +0.408 +2.482 +0.221 +135.958 +98.356 +21.002 +Musk +0.364 +0.695 +0.564 +42.303 +81.197 +12.037 +Pendigits +0.350 +1.247 +1.899 +83.944 +62.739 +15.998 +Pima +0.200 +0.018 +0.014 +10.228 +8.042 +4.247 +Vowels +0.216 +0.054 +0.031 +18.116 +14.394 +8.551 +Wbc +0.177 +0.010 +0.108 +5.136 +5.401 +5.487 +Arrhythmia +0.288 +0.025 +0.076 +7.017 +13.657 +4.157 +Cardio +0.294 +0.093 +0.275 +22.204 +17.979 +6.987 +Satellite +0.369 +1.152 +1.735 +78.876 +71.493 +13.846 +Satimage-2 +0.361 +0.916 +1.340 +72.048 +69.147 +8.580 +Shuttle +0.987 +63.146 +4.426 +594.589 +427.291 +61.471 +Thyroid +0.259 +0.278 +0.064 +46.583 +31.208 +6.856 +FMNIST +4.298 +52.114 +15.446 +744.652 +1495.926 +32.859 +Wafer +89.499 +11498.385 +910.600 +4561.028 +12363.401 +149.986 +Reuters +4.984 +106.973 +1.394 +16.567 +23.931 +92.486 + +ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models +Table A.3. AUC results of the ODIM with various number of generative models to take an ensemble. +Data +nEns = 1 +nEns = 2 +nEns = 5 +nEns = 10 +nEns = 12 +nEns = 15 +nEns = 20 +BreastW +0.988 +0.991 +0.991 +0.991 +0.991 +0.991 +0.991 +Cover +0.817 +0.863 +0.833 +0.837 +0.852 +0.868 +0.876 +Glass +0.707 +0.748 +0.750 +0.712 +0.706 +0.700 +0.707 +Ionosphere +0.846 +0.875 +0.861 +0.913 +0.907 +0.905 +0.915 +Mammography +0.825 +0.836 +0.844 +0.856 +0.857 +0.858 +0.859 +Musk +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +1.000 +Pendigits +0.885 +0.900 +0.945 +0.955 +0.954 +0.960 +0.962 +Pima +0.686 +0.690 +0.699 +0.703 +0.704 +0.705 +0.705 +Vowels +0.877 +0.882 +0.903 +0.904 +0.896 +0.895 +0.899 +Wbc +0.922 +0.922 +0.931 +0.936 +0.935 +0.938 +0.937 +Arrhythmia +0.778 +0.780 +0.783 +0.786 +0.786 +0.785 +0.783 +Cardio +0.898 +0.896 +0.915 +0.918 +0.911 +0.924 +0.920 +Satellite +0.699 +0.701 +0.693 +0.695 +0.696 +0.694 +0.693 +Satimage-2 +0.992 +0.992 +0.991 +0.992 +0.992 +0.992 +0.992 +Shuttle +0.938 +0.968 +0.979 +0.978 +0.978 +0.979 +0.979 +Thyroid +0.876 +0.913 +0.924 +0.927 +0.927 +0.926 +0.926 + diff --git a/KtE3T4oBgHgl3EQfAgkC/content/tmp_files/load_file.txt b/KtE3T4oBgHgl3EQfAgkC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d791701747a8caab724bd82ed132605e89517211 --- /dev/null +++ b/KtE3T4oBgHgl3EQfAgkC/content/tmp_files/load_file.txt @@ -0,0 +1,2212 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfAgkC/content/2301.04257v1.pdf,len=2211 +page_content='ODIM: an efficient method to detect outliers via inlier-memorization effect of deep generative models Dongha Kim 1 Jaesung Hwang 2 Jongjin Lee 3 Kunwoong Kim 3 Yongdai Kim 3 Abstract Identifying whether a given sample is an outlier or not is an important issue in various real-world domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfAgkC/content/2301.04257v1.pdf'} +page_content=' This study aims to solve the unsuper- vised outlier detection problem where training data contain outliers, but any label information about inliers and outliers is not given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfAgkC/content/2301.04257v1.pdf'} +page_content=' We pro- pose a powerful and efficient learning framework to identify outliers in a training data set using deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfAgkC/content/2301.04257v1.pdf'} +page_content=' We start with a new obser- vation called the inlier-memorization (IM) effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfAgkC/content/2301.04257v1.pdf'} +page_content=' When we train a deep generative model with data contaminated with outliers, the model first memo- rizes inliers before outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfAgkC/content/2301.04257v1.pdf'} +page_content=' Exploiting this find- ing, we develop a new method called the outlier detection via the IM effect (ODIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfAgkC/content/2301.04257v1.pdf'} +page_content=' The ODIM only requires a few updates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfAgkC/content/2301.04257v1.pdf'} +page_content=' thus, it is computa- tionally efficient, tens of times faster than other deep-learning-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfAgkC/content/2301.04257v1.pdf'} +page_content=' Also, the ODIM filters out outliers successfully, regardless of the types of data, such as tabular, image, and sequen- tial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfAgkC/content/2301.04257v1.pdf'} +page_content=' We empirically demonstrate the superiority and efficiency of the ODIM by analyzing 20 data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfAgkC/content/2301.04257v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfAgkC/content/2301.04257v1.pdf'} +page_content=' Introduction Outlier (also anomaly) is an observation that differs signifi- cantly from other observations, and outlier detection (OD) is the task of identifying outliers in a given data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfAgkC/content/2301.04257v1.pdf'} +page_content=' OD has wide applications such as fraud detection, fault detec- tion, and defect detection in images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfAgkC/content/2301.04257v1.pdf'} +page_content=' OD is also used as a pre-processing step in supervised learning to filter out anomalous training samples, which may degrade the perfor- mance of a predictive model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfAgkC/content/2301.04257v1.pdf'} +page_content=' OD problems can be categorized into three areas in general: 1) Supervised outlier detection (SOD) requires label infor- 1Department of Statistics, Sungshin Women’s University 2SK Telecom 3Department of Statistics, Seoul National University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KtE3T4oBgHgl3EQfAgkC/content/2301.04257v1.pdf'} +page_content=' Cor- respondence to: Yongdai Kim 3.0.CO;2-X. +32. Erdem R., Keskin M., Phys. Rev. E, 2001, 64, 026102, doi:10.1103/PhysRevE.64.026102. +33. Keskin M., Erdem R., Phys. Lett. A, 2002, 297, 427, doi:10.1016/s0375-9601(02)00264-5. +34. Erdem R., Ekiz C., Keskin M., Phys. Status Solidi B, 2003, 240, 220, doi:10.1002/pssb.200301876. +35. Gülpınar G., Demirhan D., Büyükkılıç F., Phys. Rev. E, 2007, 75, 021104, doi:10.1103/PhysRevE.75.021104. +36. Onsager L., Phys. Rev., 1931, 37, 405, doi:10.1103/PhysRev.37.405. +37. Onsager L., Phys. Rev., 1931, 38, 2265, doi:10.1103/PhysRev.38.2265. +38. Pawlak A., Erdem R., Gülpınar G., J. Magn. Magn. Mater., 2019, 472, 86, doi:10.1016/j.jmmm.2018.09.115. +Часи повiльної та швидкої релаксацiї квантової ґраткової +моделi з локальними багатомiнiмумними потенцiалами: +феноменологiчна динамiка в сегнетоелектричних +кристалах Sn2P2S6 +Р. Ердем1, С. Озюм2, Н. Гюджлюр3 +1 Фiзичний факультет, Унiверситет Акденiз, 07058, Анталiя, Туреччина +2 Професiйна школа Аласа Авнi Джелiк, Унiверситет Гiтiт, 19600, Чорум, Туреччина +3 Факультет фiзичної освiти, Унiверситет Неджметтина Ербакана, 42090, Конья, Туреччина +У продовження опублiкованої ранiше роботи [Velychko O. V., Stasyuk I. V., Phase Transitions, 2019, 92, 420], +для деформованої Sn2P2S6 сегнетоелектричної ґратки наведено феноменологiчну схему для розгляду ре- +лаксацiйної динамiки квантової ґраткової моделi з багатомiнiмумними потенцiалами. Схема базується на +поєднаннi статистичної рiвноважної теорiї та необоротної термодинамiки. З метою отримання узгодже- +ного опису припускається, що дипольне впорядкування чи поляризацiю (𝜂) та об’ємну деформацiю (𝑢) +можна розглядати як потоки та сили в розумiннi теорiї Онзагера. З лiнiйних спiввiдношень мiж силами +та потоками отримано рiвняння динамiки, якi характеризуються двома часами релаксацiї (𝜏𝑆, 𝜏𝐹 ), що +описують необоротнiй процес мiж рiвноважними станами. Вивчено поведiнку 𝜏𝑆 i 𝜏𝐹 поблизу сегнето- +електричних фазових переходiв. +Ключовi слова: квантова ґраткова модель, сегнетоелектричнi кристали, Sn2P2S6, часи релаксацiї, теорiя +Онзагера +43707-8 + diff --git a/L9AzT4oBgHgl3EQfkf2o/content/tmp_files/load_file.txt b/L9AzT4oBgHgl3EQfkf2o/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ae5e9c4df15ac8092255cb2e5cc51b24d304e513 --- /dev/null +++ b/L9AzT4oBgHgl3EQfkf2o/content/tmp_files/load_file.txt @@ -0,0 +1,674 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf,len=673 +page_content='Condensed Matter Physics, 2022, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 25, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 4, 43707: 1–8 DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='5488/CMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='43707 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='icmp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='lviv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='ua/journal Slow and fast relaxation times of quantum lattice model with local multi-well potentials: phenomenological dynamics for Sn2P2S6 ferroelectric crystals R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Erdem 1∗, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Özüm 2, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Güçlü 3 1 Department of Physics, Akdeniz University, 07058, Antalya, Türkiye 2 Alaca Avni Çelik Vocational School, Hitit University, 19600, Çorum, Türkiye 3 Department of Physics Education, Necmettin Erbakan University, 42090, Konya, Türkiye Received June 29, 2022, in final form October 16, 2022 As a continuation of the previously published work [Velychko O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', Stasyuk I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', Phase Transitions, 2019, 92, 420], a phenomenological framework for the relaxation dynamics of quantum lattice model with multi-well potentials is given in the case of deformed Sn2P2S6 ferroelectric lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' The framework is based on the combination of statistical equilibrium theory and irreversible thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' In order to study these dynamics in a connected way we assume that the dipole ordering or polarization (𝜂) and volume deformation (𝑢) can be treated as fluxes and forces in the sense of Onsager theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' From the linear relations between the forces and fluxes, the rate equations are derived and characterized by two relaxation times (𝜏𝑆, 𝜏𝐹 ) which describe the irreversible process near the equilibrium states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' The behaviors of 𝜏𝑆 and 𝜏𝐹 in the vicinity of ferroelectric phase transitions are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Key words: quantum lattice model, ferroelectric crystals, Sn2P2S6, relaxation times, Onsager theory 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Indroduction Sn2P2S6 (SPS) ferroelectrics (FEs) is one of the prospective materials which presents spontaneous polarization determined by anharmonic potentials for the order parameter fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Intensive ex- perimental and theoretical efforts have been devoted to the study of SPS crystals [1–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Among the theories, quantum lattice model (QLM) is of special interest for the SPS investigations [17–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' It is a lattice system with three-well local potentials related to Blume-Emery-Griffiths model [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Based on the QLM, the thermodynamics of the deformed SPS crystals has recently been studied in the presence of external pressure by Velychko and Stasyuk [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' For the monoclinic SPS-FEs, they obtained a set of mean-field self-consistent equations which describe stable states at a given mechanical stress and determined self-consistency parameters using the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' From the numerical solutions, the first- and second-order phase transitions between the ferroelectric (F) and paraelectric (P) phases were obtained as well as the tricritical point (TCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Although a few investigations are devoted to the statics of the QLM-based SPS crystals up till now, there has been no study to observe the relaxation dynamics of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' In this work, we have made use of a simple phenomenological way to define the slow and fast relaxation times near the ferroelectric phase transitions in the SPS crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Such a study of relaxation dynamics has already been presented to observe the time dependent behaviours near the phase transitions and critical phenomena in various systems [27–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' ∗Corresponding author: rerdem@akdeniz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='tr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' This work is licensed under a Creative Commons Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='0 International License.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 43707-1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='01533v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='stat-mech] 4 Jan 2023 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Erdem, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Özüm, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Güçlü In this study, giving a brief description of the QLM with local multi-well potentials and its phase transitions for Sn2P2S6 crystals under the MFA and being motivated by [31–35], we have focused on the simple relaxation study of the system using phenomenological approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Particularly, we observe the dynamic effects appearing in the regions of first- and second-order phase transitions as well as in the vicinity of the tricritical point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' The large positive values of relaxation time near the F-P phase transitions once again confirm the divergence singularity at the continuous phase transition observed in other cooperative phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' The model description and static properties The quantum lattice model with multi-well potentials is simply described by the Hamiltonian [24] � 𝐻 = ∑︁ 𝑖 � 𝐻𝑖 + � 𝐻1 + � 𝐻2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='1) where the first, second and third terms are known as the single-site, the interaction and the deformation energy parts, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Since these terms are explicitly given in [24] and [25], we skip the details here and proceed with mentioning the properties of the system at equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' These are determined self- consistently using Gibbs free energy calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Letting,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝑁,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝐽,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝜂,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝜈,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝑐0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝑢,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝜃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝜀,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝐷 and 𝜎𝑆 be the number of lattice points,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' the effective field acting on dipoles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' dipole ordering parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' volume related with one formula unit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' volume elastic constant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' deformation (or relative volume change),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' reduced temperature,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' the renormalization of the energy gap due to the deformation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' the constant of an electron-deformational interaction and mechanical stress,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' the mean-field Gibbs free energy per site is given by the following formula [24] 𝑓MF = 𝐺MF 𝑁 = 1 2𝐽𝜂2 + 1 2𝜈𝑐0𝑢2 − 𝜃 ln � 1 + 2 exp � −𝜀 + 𝐷𝑢 𝜃 � cosh � 𝐽𝜂 2𝜃 �� − 𝜈𝑢𝜎𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='2) Here, we define 𝐽 = � 𝑗 𝐽𝑖 𝑗, 𝜂 = ⟨𝑠𝑖⟩ (𝑠𝑖 variable related to the local dipole moment, ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='⟩ denotes the thermal expectation value), 𝜃 = 𝑘𝑇 (𝑘 denotes the Boltzmann constant, 𝑇 means the absolute temperature), 𝜎𝑆 = −𝑝 (𝑝 is the hydrostatic pressure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' The minimization of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='2) with respect to the variables 𝜂 and 𝑢 are treated as two independent variational parameters and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='2) in the MFT give the following conditions 𝜕 𝑓MF 𝜕𝜂 = 0, 𝜕 𝑓MF 𝜕𝑢 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='3) For the system at equilibrium, the self-consistent equations are expressed in the form 𝜂 = e−𝑦𝑧 1 + 2e−𝑦𝑥 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='4) 𝜎 = 𝑢𝑐0 + 𝐷 𝜈 � 2e−𝑦𝑥 1 + 2e−𝑦𝑥 � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='5) where 𝑥 = cosh � 𝐽 𝜂 2𝜃 � , 𝑦 = 𝜀+𝐷𝑢 𝜃 , 𝑧 = sinh � 𝐽 𝜂 2𝜃 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' In order to have an insight into the phase transitions undergoing in the SPS-FEs on the monoclinic lattice structure, Velychko and Stasyuk [24] firstly solved the above equations using the parameter values of 𝐽 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='14 eV, 𝜈 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='23 × 10−21 cm3, 𝑐0 = 5 × 1011 erg/cm3, 𝜈𝑐0 = 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='8 eV, 𝐷 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='1 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' They obtained the polarization 𝜂 and deformation 𝑢 as functions of the energy gap 𝜀 and pressure 𝑝 at different temperature values 𝜃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Recently, Erdem [25] improved their results by adding 𝜂 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝜃 and 𝑢 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝜃 at a fixed pressure to represent the effect of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Here, we summarize the basic results from these references for the convenience of our later discussions as follows: a finite jump of 𝜂 and 𝑢 at the first-order (discontinuous) phase transition from F phase to P phase accompanied by compression of the lattice occured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' These jumps reduce as the TCP is approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Above this point the phase transition becomes of the second-order (continuous), and thus the jumps of the parameters 𝜂 and 𝑢 vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 43707-2 Slow and fast relaxation times of quantum lattice model with local multi-well potentials 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Theoretical framework In order to study the relaxation dynamics of the above system we now assume that a small external stimulation is applied removing the system slightly from equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' The Gibbs free energy produced in the irreversible process (Δ 𝑓MF) is calculated to observe how rapidly the system relaxes back to equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' In this case, the Gibbs free energy near the equilibrium states will be 𝑓MF(𝑇, 𝜂, 𝑢) = 𝑓 (0) MF (𝑇, 𝜂0, 𝑢0) + Δ 𝑓MF, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='1) where 𝑓 (0) MF (𝑇, 𝜂0, 𝑢0) is the equilibrium free energy [found from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='2) by setting 𝜂 = 𝜂0, 𝑢 = 𝑢0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Δ 𝑓MF is given by the terms in a Taylor series expansion of 𝑓MF with respect to the spontaneous equilibrium point 𝜂 = 𝜂0, 𝑢 = 𝑢0 as Δ 𝑓MF = 1 2 � 𝜙𝜂𝜂 (𝜂 − 𝜂0)2 + 2𝜙𝜂𝑢 (𝜂 − 𝜂0) (𝑢 − 𝑢0) + 𝜙𝑢𝑢 (𝑢 − 𝑢0)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='2) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='2), the expressions given below for 𝜙𝜂𝜂, 𝜙𝜂𝑢, 𝜙𝑢𝑢 are explicit functions of the known equilibrium quantities 𝜂0, 𝑢0 and take the form 𝜙𝜂𝜂 = � 𝜕2 𝑓MF 𝜕𝜂2 � eq , 𝜙𝜂𝑢 = � 𝜕2 𝑓MF 𝜕𝜂𝜕𝑢 � eq = 𝜙𝑢𝜂 = � 𝜕2 𝑓MF 𝜕𝑢𝜕𝜂 � eq , 𝜙𝑢𝑢 = � 𝜕2 𝑓MF 𝜕𝑢2 � eq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='3) Here, the subscribe ‘eq’ means equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' The generalized forces (𝑋𝜂, 𝑋𝑢) can be described using the derivatives with respect to deviations (𝜂 − 𝜂0, 𝑢 − 𝑢0), respectively: 𝑋𝜂 = − 𝜕(Δ 𝑓MF) 𝜕(𝜂 − 𝜂0) = −𝜙𝜂𝜂 (𝜂 − 𝜂0) − 𝜙𝜂𝑢 (𝑢 − 𝑢0) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='4) 𝑋𝑢 = − 𝜕(Δ 𝑓MF) 𝜕(𝑢 − 𝑢0) = −𝜙𝑢𝜂 (𝜂 − 𝜂0) − 𝜙𝑢𝑢 (𝑢 − 𝑢0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='5) Based on the Onsager reciprocity theorem (ORT) [36, 37], a linear relation between the currents ( �𝜂, �𝑢) and forces (𝑋𝜂, 𝑋𝑢) is introduced as follows in terms of a matrix of phenomenological rate coefficients (or Onsager constants): � �𝜂 �𝑢 � = − � 𝛾𝜂 𝛾 𝛾 𝛾𝑢 � � 𝑋𝜂 𝑋𝑢 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='6) where the corresponding off-diagonal elements on the two sides of the main diagonal have the same signs (the matrix is symmetric since both 𝜂 and 𝑢 are even variables under time inversion), and the dot denotes a derivative with respect to time 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' The above matrix equation yields, upon using equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='5), the rate equations: �𝜂 = d𝜂 d𝑡 = −Φ𝜂𝜂 (𝜂 − 𝜂0) − Φ𝜂𝑢 (𝑢 − 𝑢0) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='7) �𝑢 = d𝑢 d𝑡 = −Φ𝑢𝜂 (𝜂 − 𝜂0) − Φ𝑢𝑢 (𝑢 − 𝑢0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='8) The coefficients are defined by: Φ𝜂𝜂 = 𝛾𝜂𝜙𝜂𝜂+𝛾𝜙𝜂𝑢, Φ𝜂𝑢 = 𝛾𝜂𝜙𝜂𝑢+𝛾𝜙𝑢𝑢, Φ𝑢𝜂 = 𝛾𝜙𝜂𝜂+𝛾𝑢𝜙𝜂𝑢, Φ𝑢𝑢 = 𝛾𝑢𝜙𝑢𝑢+𝛾𝜙𝜂𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='9) Assuming a solution of the form exp(−𝑡/𝜏) for equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='8), one obtains the secular equation ���� 𝜏−1 − Φ𝜂𝜂 −Φ𝜂𝑢 −Φ𝑢𝜂 𝜏−1 − Φ𝑢𝑢 ���� = 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='10) 43707-3 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Erdem, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Özüm, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Güçlü which yields two inverse relaxation times [38]: 1 𝜏𝑆 = 1 2 (Φ𝜂𝜂 + Φ𝑢𝑢) − 1 2 � (Φ𝜂𝜂 − Φ𝑢𝑢)2 + 4Φ𝜂𝑢Φ𝑢𝜂 �1/2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='11) 1 𝜏𝐹 = 1 2 (Φ𝜂𝜂 + Φ𝑢𝑢) + 1 2 � (Φ𝜂𝜂 − Φ𝑢𝑢)2 + 4Φ𝜂𝑢Φ𝑢𝜂 �1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='12) Here, 𝜏𝑆 corresponds to a slower relaxation process while 𝜏𝐹 corresponds to the faster one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' They characterize the dipole ordering parameter and deformation (or relative volume change) relaxations, respectively, in the phenomenological approach using the ORT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Numerical results Firstly, the behavior of slow (𝜏𝑆) and fast (𝜏𝐹) relaxation times [in seconds (𝑠) unit] as a function of the energy gap 𝜀 (in electronvolts (eV) unit) is shown in figure 1 for several values of 𝜃 when there is no external pressure 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' For Onsager rate coefficients, we choose 𝛾𝜂 = 𝛾𝑢 = 1 eV−1s−1, 𝛾 = 10−5 eV−1s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' The temperature values on the figures are for the second-order phase transition, the first-order phase transition and TCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' The vertical dotted lines refer to the phase transition values of 𝜀, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', 𝜀𝐶, 𝜀𝐷 and 𝜀TCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' In this case, 𝜏𝑆 increases rapidly while rising (lowering) 𝜀 value on left (right) side of 𝜀𝐶/𝜀TCP and diverges to infinity, illustrated by the blue/red curves in figure 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' On the contrary, it presents a large and abrupt drop across the discontinuous phase transition point [see figure 1(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Former singularity in 𝜏𝑆 is an expected property known as the signature of continuous phase transitions and hence well agrees with earlier investigations [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' However, the latter one is a novel property which is not found before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' As for the fast relaxation time 𝜏𝐹, it also increases with 𝜀 but remains scarcerly varied just around the continuous phase transition (tricritical point), as also shown via blue (red) curves in figure 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Hence, both critical and tricritical behaviours of 𝜏𝐹 are different from 𝜏𝑆, just a cusp singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Beyond 𝜀𝐶, a maximum of 𝜏𝐹 is seen but it disappears at 𝜀TCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' It should be stressed that Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' (Colour online) (a), (b) Slow relaxation time 𝜏𝑆 and (c), (d) fast relaxation time 𝜏𝐹 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' energy gap 𝜀 at 𝜃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='0230, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='0179, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='0151 eV for 𝛾𝜂 = 𝛾𝑢 = 1 eV−1s−1, 𝛾 = 10−5 eV−1s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 43707-4 Slow and fast relaxation times of quantum lattice model with local multi-well potentials Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' (Colour online) (a), (b) Slow relaxation time 𝜏𝑆 and (c), (d) fast relaxation time 𝜏𝐹 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' energy gap 𝜀 in the absence and presence of the off-diagonal rate coefficient when 𝑝 = 0, 𝜃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='0230 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' (Colour online) (a), (b) Slow relaxation time 𝜏𝑆 and (c), (d) fast relaxation time 𝜏𝐹 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' pressure 𝑝 at 𝜃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='0225, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='0175, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='0125 𝑒𝑉 for 𝛾𝜂 = 𝛾𝑢 = 1 eV−1s−1, 𝛾 = 10−5 eV−1s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' cusps also occurred for 𝜏𝐹 during the discontinuous phase transition, indicated in figure 1(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' These findings are also in agreement with previous relaxation studies [32, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' In order to show the effects of rate constants on the relaxation process, we illustrate in figure 2 the 43707-5 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Erdem, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Özüm, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Güçlü Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' (Colour online) (a), (b) Slow relaxation time 𝜏𝑆 and (c), (d) fast relaxation time 𝜏𝐹 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' temperature 𝜃 in the absence and in the presence of the off-diagonal Onsager coefficient for 𝜀 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='011 eV, 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='5 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' energy gap 𝜀 variation of 𝜏𝑆 and 𝜏𝐹 in the absence and in the presence of the off-diagonal Onsager constant for 𝑝 = 0 and 𝜃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='0230 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' As seen in figure 2(a), on both sides of 𝜀𝐶, illustrated by a vertical dotted line, the divergence of 𝜏𝑆 gets pushed away from the critical energy gap value as we decrease the values of 𝛾𝜂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' This means that the rising of 𝛾𝜂 leads to a speed up of the whole relaxation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' The opposite is valid in the case of off-diagonal Onsager coefficient 𝛾 in figure2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝜏𝐹 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝜀 is shown for different values of 𝛾𝜂 and 𝛾 for 𝑝 = 0 and 𝜃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='0230 eV in figures 2(c) and 2(d), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' It is evident that 𝜏𝐹 curves are independent of 𝛾𝜂 and 𝛾 values, shown in figures 2(c) and 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Next, 𝜏𝑆 and 𝜏𝐹 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' pressure 𝑝 [in gigapascal (GPa) unit] is displayed in figure 3 for three different temperatures using 𝛾𝜂 = 𝛾𝑢 = 1 eV−1s−1, 𝛾 = 10−5 eV−1s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Figure3 also displays the pressure dependence of 𝜏𝑆 and 𝜏𝐹 where we adjust the parameters so as to obtain an agreement with figures 6(a), 6(b) in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Our calculated dependence of (𝜏𝑆, 𝜏𝐹) on 𝑝 demonstrates a good correspondence to previous 𝜀 dependence of (𝜏𝑆, 𝜏𝐹) in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Our plots of the critical and tricritical cases are presented in figures 3(a), 3(c) (blue and red curves) and the first-order case in figures 3(b), 3(d) (green curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' In other words, the overall critical and tricritical behaviours of (𝜏𝑆, 𝜏𝐹) do not change so much at the given 𝜀 and 𝜃 values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' However, the amount of the finite jump in 𝜏𝑆 at the first-order phase transition point seen in figure 3(b) is very much smaller than that of figure 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' It is important to mention that, because of the similar behavior of figure 2, the pressure 𝑝 variation of 𝜏𝑆 and 𝜏𝐹 in the absence and in the presence of 𝛾 for 𝑝 = 0 and 𝜃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='0230 eV has not been drawn here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Finally, the dependence of 𝜏𝑆 and 𝜏𝐹 on the temperature 𝜃 can be followed in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' These quantities are plotted for both 𝛾 = 0 and 𝛾 ≠ 0 characterizing the speed of the relaxation process when 𝜀 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='011 eV, 𝑝 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='5 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' It is clearly visible that the increases of 𝛾𝜂(𝛾) result in much faster (slower) increase of 𝜏𝑆 around the critical temperature during the critical slowing down process [figures 4(a) and 4(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' These are in direct agreement with previous results gained from dynamic studies for similar physical systems [31–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' It is quite evident from figures 4(c) and 4(d) that the investigated plots of 𝜏𝐹 display a cusp-singularity at the critical temperature regardless of the value of rate parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' By comparison of the colored curves in figures 4(c) and 4(d), it can be stated that 𝜏𝐹 is not sensitive to any change in 𝛾𝜂 when 𝛾 = 0 and in 𝛾 for 𝛾 ≠ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 43707-6 Slow and fast relaxation times of quantum lattice model with local multi-well potentials 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Conclusions In this work, we have studied the near equilibrium relaxation dynamics of the QLM for the ferro- electric SPS crystals by means of ORT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' More particularly, time derivatives of polarization 𝜂 and volume deformation 𝑢 are treated as fluxes or currents conjugate to their appropriate forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' The forces are found using the mean-field free energy production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' A set of rate (or kinetic) equations are derived from the linear relation between the forces and currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' In terms of the phenomenological constants, the solutions of rate equations are expressed as a set of two relaxations times denoted by 𝜏𝑆 and 𝜏𝐹.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Using the parameter values related to the SPS crystals in [24, 25], we have plotted these two quantities as a function of the energy gap, external pressure and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Although the present calculations on 𝜏𝑆 and 𝜏𝐹 display exactly the same critical and tricritical behaviors as in the earlier investigations, we observe unusual results during the discontinous phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' In other words, the slow relaxation time 𝜏𝑆 shows a large jump in the 𝜏𝑆 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝜀 and 𝜏𝑆 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝑝 plots while fast relaxation time 𝜏𝐹 has a cusp-singularity and small jump in the 𝜏𝐹 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝜀 and 𝜏𝐹 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 𝑝 plots, respectively, at the first-order phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' In particular, comparing figures 1(b) and 1(d) with corresponding plots in [32] it can be stated that this qualitative difference distinguishes our results from the earlier ORT calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Eijt S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' W.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', Stoika I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', Vysochanskii Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Alloys Compd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', 2017, 694, 808, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='750.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Yonemitsu K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' B, 2008, 78, 205102, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', Stasyuk I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', Phase Transitions, 2019, 92, 420, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='1080/01411594.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='1582051.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Blume M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', Emery V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', Griffiths R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' A, 1971, 4, 1071, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='1103/PhysRevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='1071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Tanaka 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', 1962, 37, 1397, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='1733295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Barry J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' H.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Barry J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', Harrington D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' B, 1971, 4, 3068, 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', 1977, 43, 1832, doi:https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='1143/JPSJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} 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M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' E, 2001, 64, 026102, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='1103/PhysRevE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content='026102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} 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+page_content='115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Часи повiльної та швидкої релаксацiї квантової ґраткової моделi з локальними багатомiнiмумними потенцiалами: феноменологiчна динамiка в сегнетоелектричних кристалах Sn2P2S6 Р.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Ердем1, С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Озюм2, Н.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Гюджлюр3 1 Фiзичний факультет, Унiверситет Акденiз, 07058, Анталiя, Туреччина 2 Професiйна школа Аласа Авнi Джелiк, Унiверситет Гiтiт, 19600, Чорум, Туреччина 3 Факультет фiзичної освiти, Унiверситет Неджметтина Ербакана, 42090, Конья, Туреччина У продовження опублiкованої ранiше роботи [Velychko O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', Stasyuk I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=', Phase Transitions, 2019, 92, 420], для деформованої Sn2P2S6 сегнетоелектричної ґратки наведено феноменологiчну схему для розгляду ре- лаксацiйної динамiки квантової ґраткової моделi з багатомiнiмумними потенцiалами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Схема базується на поєднаннi статистичної рiвноважної теорiї та необоротної термодинамiки.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' З метою отримання узгодже- ного опису припускається, що дипольне впорядкування чи поляризацiю (𝜂) та об’ємну деформацiю (𝑢) можна розглядати як потоки та сили в розумiннi теорiї Онзагера.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' З лiнiйних спiввiдношень мiж силами та потоками отримано рiвняння динамiки, якi характеризуються двома часами релаксацiї (𝜏𝑆, 𝜏𝐹 ), що описують необоротнiй процес мiж рiвноважними станами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Вивчено поведiнку 𝜏𝑆 i 𝜏𝐹 поблизу сегнето- електричних фазових переходiв.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} +page_content=' Ключовi слова: квантова ґраткова модель, сегнетоелектричнi кристали, Sn2P2S6, часи релаксацiї, теорiя Онзагера 43707-8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfkf2o/content/2301.01533v1.pdf'} diff --git a/MdAyT4oBgHgl3EQfs_ko/content/tmp_files/2301.00585v1.pdf.txt b/MdAyT4oBgHgl3EQfs_ko/content/tmp_files/2301.00585v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5c8a67fc151d75f5efc97fb1178e7df0894412a9 --- /dev/null +++ b/MdAyT4oBgHgl3EQfs_ko/content/tmp_files/2301.00585v1.pdf.txt @@ -0,0 +1,1903 @@ +arXiv:2301.00585v1 [math.AP] 2 Jan 2023 +INVERSE SOURCE PROBLEM FOR THE PSEUDO-PARABOLIC +EQUATION ASSOCIATED WITH THE JACOBI OPERATOR +BAYAN BEKBOLAT AND NIYAZ TOKMAGAMBETOV +Abstract. In this paper we investigate direct and inverse problems for time- +fractional pseudo-parabolic equations associated with the Jacobi operator. +The +existence and uniqueness of the solutions are proved. Also, the stability result of +the inverse source problem (ISP) is established. +1. Introduction +The main object of this paper is the following non-homogeneous time-fractional +pseudo-parabolic equation on the domain D = {(t, x) : 0 < t < T < ∞, x ∈ R+ = +(0, ∞)} +Dγ +0+,t (u(t, x) − a∆α,βu(t, x)) − ∆α,βu(t, x) + mu(t, x) = f(x), +where 0 < γ ≤ 1, with non–negative constants m and a, and with the initial condition +u(0, x) = φ(x), +x ∈ R+, +where Dγ +0+,t is given by +Dγ +0+,t = +� +Dγ +0+,t, +0 < γ < 1, +d +dt, +γ = 1, +Dγ +0+,t is the left-sided Caputo fractional derivative and ∆α,β is the Jacobi operator +given by the expression +(1.1) +∆α,β = A−1 +α,β(x) d +dx +� +Aα,β(x) d +dx +� +, +x ∈ (0, ∞). +Here, we denote by Aα,β(x) = 22ρ(sinh(x))2α+1(cosh(x))2β+1, ρ = α + β + 1, with +α ≥ β ≥ −1 +2. +In our studies we would be questioned about the well–posedness of the direct prob- +lem and the stability of the inverse source problem with the additional information +– over-determination condition +u(T, x) = ψ(x), +x ∈ R+. +For the ISP we will restore the pair (u, f) under some conditions on the function ψ. +Date: January 3, 2023. +2020 Mathematics Subject Classification. Primary 35R30; Secondary 35R11, 35C15. +Key words and phrases. Jacobi operator, Jacobi transform, time-fractional pseudo-parabolic +equation, inverse source problem. +This research was funded by the Science Committee of the Ministry of Science and Higher Edu- +cation of the Republic of Kazakhstan (Grant No. AP14972634). +1 + +2 +B. BEKBOLAT AND N. TOKMAGAMBETOV +One of the first mathematicians who studied the ISP was Rundell [Run80] in 1980s. +He considered the evolution type equation +(1.2) +du +dt + Au = f +in a Banach space X, where A is linear operator in X and f is a constant vector in +X, with conditions +u(0) = u0, +and +u(T) = u1. +Using semigroups of operators Rundell proved a general theorem about the existence +of a unique solution pair (u(t), f) of the problem, which then was applied to equa- +tions of parabolic and pseudo-parabolic types. When the non-homogeneous term is +represented in the form f(t) = Φ(t)f, where Φ(t) is known operator and the element +f is unknown, and A is a closed linear operator from Lp(Ω) into Lp(Ω), several ISPs +for the equation (1.2) were studied by A.I. Prilepko and I.V. Tikhonov [PT92] in +1992. They applied obtained results to the transport equation. In the general case, +where the unknown source depends on time, under a sufficient condition, ISPs for +the equation (1.2) with the linear elliptic partial differential operator A of order 2m +with the bounded measurable coefficients such that +(Aϕ, ϕ) ≥ ∥ϕ∥2 +for all ϕ ∈ H2m(Ω) ∩ Hm +0 (Ω), µ = constant > 0 was investigated by I. Bushuyev +[Bus95] in 1995. +Nonetheless, there is no general closed theory for abstract case of F(x, t). Known +results deal with separated source terms. In 2002 I.V. Tikhonov and Yu.S. Eidelman +[TE02] considered ISPs for the generalization of the equation (1.2) of the form +dNu(t) +dtN += Au(t) + p, +0 < t < T, +for some positive integer N ≥ 1 and some real number T > 0 with an unknown +parameter p and a closed linear operator A in the Banach space under the Cauchy +conditions and ”over-determination condition” u(T) = uN (also in the Banach space). +For the Laplace operator (−∆) which is one of the most interesting examples in +Physics, M. Choulli and M. Yamamoto in [CY04] established the uniqueness and +conditional stability in determining a heat source term from boundary measurements +with f = σ(t)ϕ(x), where σ(t) is known. +Asymptotic behaviour of the solution of the inverse source problem for the pseudo- +parabolic equation +(u(x, t) − ∆u(x, t))t − ∆u(x, t) + αu(x, t) = f(t)g(x, t), +Q∞ = Ω × (0, ∞) +with a integral over-determination condition was studied by M. Yaman and ¨O. F. +G¨oz¨ukızıl in [YG03] in 2004. +Fractional derivatives and fractional partial differential equations have received +great attention both in analysis and application, which are used in modeling several +phenomena in different areas of science such as biology, physics, and chemistry, so the +fractional computation is increasingly attracted to mathematicians in the last several +decades. ISP for the time fractional parabolic equation +cDα +t u(x, t) = rα(Lu)(x, t) + f(x)h(x, t), +x ∈ Ω, t ∈ (0, T), 0 < α < 1, + +PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR +3 +where cDα +t is the Caputo derivative defined by +cDα +t g(t) = +1 +Γ(1 − α) +� t +0 +(t − τ)−α d +dτ g(τ)dτ +and L is a symmetric uniformly elliptic operator was considered by K. Sakamoto +and M. Yamamoto in [SY11] in 2011. The authors proved that the inverse problem +is well-posed in the Hadamard sense except for a discrete set of values of diffusion +constants using final overdetermining data. Blow-up solution and stability to ISP for +the pseudo-parabolic equation +ut − a∆ut − ∆u + +n +� +i=1 +biuxi − |u|pu = f(t)g(t), +x ∈ Ω, t > 0 +with the integral overdetermination condition was studied by Metin Yaman in [Yam12] +in 2012. ISP for the equation (1.2) considered by M.M. Slodi˘cka in [Slo13] in 2013, +when A is a linear differential operator of second-order, strongly elliptic, and the right- +hand side f is assumed to be separable in both variables x and t, i.e. f(x, t) = g(x)h(t) +(in this case h(t) is unknown). ISP for a semilinear time-fractional diffusion equation +of second order in a bounded domain in Rd +(g1−β ∗ ∂tu(x))(t) + L(x, t)u(x, t) = h(t)f(x) + +� t +0 +F(x, s, u(x, s))ds +with a linear second order differential operator L(x, t) in the divergence form with +space and time dependent coefficients was studied by M. Slodi˘cka and M. ˘Si˘skova +in [SS16] in 2016. +Authors showed the existence, uniqueness and regularity of a +weak solution (u, h) ([SS16, Theorem 2.1, p. 1658]). One of the recent papers for +inverse source problems for pseudo-parabolic equations with fractional derivatives +is [RSTT21] (in 2021). In [RSTT21], authors have considered solvability of an in- +verse source problem for the pseudo-parabolic equation with the Caputo fractional +derivative Dα +t of order 0 < α ≤ 1 +Dα +t (u(t) + Lu(t)) + Mu(t) = f(t) +in +H, +u(0) = φ ∈ H, +u(T) = ψ ∈ H, +where H be a separable Hilbert space and L, M be operators with the corresponding +discrete spectra on H. The authors obtained well-posedness results. +A number of articles address the solvability of the inverse problems for the diffusion +and sub-diffusion equations ([CNYY09, JR15, KS10, KST17, OS12a, OS12b, RTT19]) +and fractional diffusion equations ([SSB19, TT17, WYH13]). +The semigroups (H(α,β) +t +)t≥0 (the solution of the heat equation associated with the +Jacobi-Dunkl operator Λ2 +α,β ) generate a new family of Markov processes on the real +line. On some Riemannian symmetric spaces this process is the radial part of the +Brownian motion for particular values of (α, β) [CGM06]. +However, the ISP for the pseudo-parabolic equations generated by the Jacobi op- +erator ∆α,β (1.1) have not been still considered. +So, our goal is to consider the +ISP for the pseudo-parabolic equation with this special operator. Harmonic analysis +associated with the operator ∆α,β has been studied by M. Flensted-Jensen and T. +H. Koornwinder [FJ72, FJK73, FJK79, Koo75]. The spectral decomposition of the + +4 +B. BEKBOLAT AND N. TOKMAGAMBETOV +Jacobi operator was considered by M. Flensted-Jensen in 1972 [FJ72]. There were +obtained a generalization of the classical Paley-Wiener Theorem and a generalized +Fourier transform Fα,β, is called Jacobi-Fourier transform. Eigenfunctions ϕα,β +λ (x) +of the operator Jacobi is called the Jacobi function, which is hypergeometric func- +tion. The pseudo-differential operators (see [SD98]) and Sobolev type spaces Gs,p +α,β +(see [SD00]) associated with the Jacobi operator was studied by N. Ben Salem and +A. Dachraoui. In [SD98], authors proved that a pseudo-differential operator associ- +ated with a symbol in Sm +0 is a continuous linear mapping from some subspace of the +Schwartz space into itself. +Our main result reads as follows. +Theorem 1.1. Let 0 < γ ≤ 1. Assume that ψ, φ ∈ H. Then the pair (u, f) is a +unique solution of the ISP, which are functions u ∈ Cγ([0, T], L2(µ))∩C([0, T], H), f ∈ +L2(µ) can be represented by the formulas +u(t, x) = +� ∞ +0 +� ∞ +0 +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)tγ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +�ψ(y)ϕα,β +λ (y)ϕα,β +λ (x)dµα,β(y)dνα,β(λ) +− +� ∞ +0 +� ∞ +0 +Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ� +− Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)tγ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +× φ(y)ϕα,β +λ (y)ϕα,β +λ (x)dµα,β(y)dνα,β(λ) +and +f(x) = +� ∞ +0 +� ∞ +0 +(λ2 + ρ2 + m) +ψ(y) − φ(y)Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +× ϕα,β +λ (y)ϕα,β +λ (x)dµα,β(y)dνα,β(λ). +The contents of this paper as follows. In Section 2, we collect some results about +harmonic analysis associated with the Jacobi operator on R+ and here we introduce +the Sobolev type space H, also given some necessary information about fractional +derivative. In Section 3, we prove Theorem 3.1 for the direct problem. In Section +4, we prove our main Theorem 3.2 about solvability of the inverse source problem +associated with the Jacobi operator on R+, also shown stability analysis and example +for the inverse source problem. +2. Preliminaries +2.1. Jacobi analysis. The singular second order differential equation ([FJ72]) +∆α,βϕα,β +λ (x) + (λ2 + ρ2)ϕα,β +λ (x) = 0 +on +(0, ∞) +with initial conditions +ϕα,β +λ (0) = 1, +d +dtϕα,β +λ (0) = 0 + +PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR +5 +has a unique solution, given by the expression +(2.1) +ϕα,β +λ (x) = 2F1 +�1 +2(ρ + iλ), 1 +2(ρ − iλ); α + 1; − sinh2 x +� +, +where 2F1 is the Gauss hypergeometric function. The function ϕα,β +λ +(2.1) is called the +Jacobi function and analytic for x ∈ [0, ∞) and +ϕα,β +λ (x) = ϕα,β +−λ (x) +and +ϕα,β +λ (x) = ϕα,β +λ (x). +In particularly, we have +ϕ +− 1 +2,− 1 +2 +λ +(x) = cos(λx). +Remark 2.1. ([FJ72, Proposition 1, p. 144]) For each fixed x ∈ (0, ∞), as a function +of λ, ϕα,β +λ +is an entire function. +Properties of the Jacobi function: +1. For all λ ∈ C and x ∈ [0, ∞), we have ([FJ72, Lemma 11, p. 153]) +i) |ϕα,β +λ (x)| ≤ ϕα,β +iImλ(x), +ii) If |Imλ| ≥ ρ then |ϕα,β +λ (x)| ≤ e(|Imλ|−ρ)x, +iii) If |Imλ| ≤ ρ then |ϕα,β +λ (x)| ≤ 1. +2. For all n ∈ Z+ there exists Kn > 0 such that ([FJ72, Theorem 2, p. 145]) +���� +dn +dxnϕα,β +λ (x) +���� ≤ Kn(1 + x)(1 + |λ|)ne(|Imλ|−ρ)x +and +���� +dn +dλnϕα,β +λ (x) +���� ≤ Kn(1 + x)n+1e(|Imλ|−ρ)x +for all λ ∈ C, x ∈ [0, ∞). +Let us introduce the following functions spaces ([FJ72, p. 146-147], [SD98, p. 368]). +Let Se(R) be the space of even, rapidly decreasing, and C∞-functions on R, +equipped with usual Schwartz topology, and Sr +e(R) = {(cosh x) +−2ρ +r Se(R)}, 0 < r ≤ 2 +be the space with the topology defined by the semi-norms +Nn,k(f) = sup +x≥0 +(cosh x) +2ρ +r (1 + x)n +���� +dk +dxk f(x) +���� . +Clearly Sr +e(R) is invariant under ∆α,β and the semi-norms defined by +Nn,k(f) = sup +x≥0 +(cosh x) +2ρ +r (1 + x)n|∆k +α,βf(x)| +are continuous on Sr +e(R). +Let Lp(R+, µα,β), 1 ≤ p < ∞ be the space of measurable functions f on R+ such +that +∥f∥p +p,µ = +� ∞ +0 +|f(x)|pdµα,β(x) < ∞, +where dµα,β(x) = (2π)− 1 +222ρ(sinh x)2α+1(cosh x)2β+1dx or dµα,β(x) = (2π)− 1 +2Aα,β(x)dx. +Remark 2.2. [FJ72, p. 146] Notice that Sr +e(R) ⊂ Lr(R+, µα,β) for all 0 < r ≤ 2 and +if r ≤ s then Sr +e(R) ⊆ Ss +e(R) ⊂ L2(R+, µα,β). + +6 +B. BEKBOLAT AND N. TOKMAGAMBETOV +Let Lp(R+, να,β), 1 ≤ p < ∞ be the space of measurable functions g on R+ such +that +∥f∥p +p,ν = +� ∞ +0 +|g(λ)|pdνα,β(λ) < ∞, +where dνα,β(λ) = (2π)− 1 +2|cα,β(λ)|−2dλ. Here, cα,β(λ) is the Harish–Chandra function, +given by +cα,β(λ) = 2ρ−iλΓ(iλ)Γ(α + 1) +Γ( ρ+iλ +2 )Γ( α−β+1+iλ +2 +). +For short, we use notations Lp(µ) and Lp(ν) instead Lp(R+, µα,β) and Lp(R+, να,β), +respectively. +For f ∈ L1(µ) the Fourier-Jacobi transform Fα,β of f is defined by ([FJ72, Propo- +sition 3, p. 146], [SD98, Definition 1.1, p. 369]) +(2.2) +�f(λ) = (Fα,βf)(λ) = +� ∞ +0 +f(x)ϕα,β +λ (x)dµα,β(x) +and for g ∈ L1(ν) the inverse Fourier-Jacobi transform F −1 +α,β is given by +(2.3) +� +F −1 +α,βg +� +(x) = +� ∞ +0 +g(λ)ϕα,β +λ (x)dνα,β(λ), +where ϕα,β +λ +is the Jacobi functions (2.1). +Proposition 2.3. ([FJ72, p. 145-146]) The operator in L2(µ) defined by ∆α,β with +domain +D0 +α,β = {u ∈ L2(µ) : u +and +u′ +are absolutely continuous and +∆α,βu ∈ L2(µ)} +can be restricted to a domain Dα,β, such that ∆α,β becomes self-adjoint. ∆α,β contains +at least functions in D0 +α,β which are differentiable at zero. ∆α,β has limit-point at ∞; +and at zero there is limit-point if 2α + 1 ≥ 3, and limit-circle if 2α + 1 < 3. In this +last case Dα,β ̸= D0 +α,β and choosing λ1 ∈ C with Imλ2 +1 > 0 we can define +Dα,β = {u ∈ D0 +α,β : lim +x→0(Aα,β(x) · (ϕα,β +λ1 (x)u′(x) − +� d +dxϕα,β +λ1 (x) +� +u(x))) = 0}. +Proposition 2.4. ([FJ72, Proposition 3, p. 146]) For f ∈ L2(µ) and λ ∈ R+ define +�f the integral converging in L2(ν). f → �f is a linear, normpreserving map of L2(µ) +onto L2(ν), the inverse given by +f(x) = +� ∞ +0 +g(λ)ϕα,β +λ (x)dνα,β(λ) +the integral converging in L2(µ). A function f ∈ L2(µ) belongs to Dα,β if and only if +(λ2 + ρ2) �f(λ) ∈ L2(ν) and in that case +� +∆α,βf(λ) = −(λ2 + ρ2) �f(λ). +In particularly, we have for Plancherel’s identity +(2.4) +∥ �f∥2,ν = ∥f∥2,µ. + +PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR +7 +Remark 2.5. For α = β = −1 +2, we have the Fourier-cosine transform +�fc(λ) = (Fcf)(λ) = +1 +√ +2π +� ∞ +0 +cos(λx)f(x)dx, +and the inverse Fourier-cosine transform is defined by +� +F −1 +c g +� +(x) = +4 +√ +2π +� ∞ +0 +cos(λx)g(λ)dλ. +Definition 2.6. We define the space +H := {u ∈ L2(µ) : (·2 + ρ2)�u ∈ L2(ν)} +with norm +∥u∥2 +H := +� ∞ +0 +|(λ2 + ρ2)�u(λ)|2dνα,β(λ). +2.2. Fractional differentiation operators. In this subsection, we introduce frac- +tional differentiation operators and other conceptions. +Definition 2.7. [KST06, p. 69] Let [a, b] (−∞ < a < b < ∞) be a finite interval on +the real axis R. The left and right Riemann-Liouville fractional integrals Iγ +a+ and Iγ +b− +of order γ ∈ R (γ > 0) are defined by +Iγ +a+[f](t) := +1 +Γ(γ) +� t +a +(t − s)γ−1f(s)ds, +t ∈ (a, b], +and +Iγ +b−[f](t) := +1 +Γ(γ) +� b +t +(t − s)γ−1f(s)ds, +t ∈ [a, b), +respectively. Here Γ denotes the Euler gamma function. +Definition 2.8. [KST06, p. +70] The left and right Riemann-Liouville fractional +derivatives Dγ +a+ and Dγ +b− of order γ ∈ R (0 < γ < 1) are given by +Dγ +a+[f](t) := d +dtI1−γ +a+ [f](t), +∀t ∈ (a, b], +and +Dγ +b−[f](t) := − d +dtI1−γ +b− [f](t), +∀t ∈ [a, b), +respectively. +Definition 2.9. [KST06, p. 91] The left and right Caputo fractional derivatives Dγ +a+ +and Dγ +b− of order γ ∈ R (0 < γ < 1) are defined by +Dγ +a+[f](t) := Dγ +a+[f(t) − f(a)], +t ∈ (a, b], +and +Dγ +b−[f](t) := Dγ +b−[f(t) − f(b)], +t ∈ [a, b), +respectively. +Definition 2.10. [CF18, p. 18, Definition 3] Let X be a Banach space. We say that +u ∈ Cγ([0, T], X) if u ∈ C([0, T], X) and Dγ +t u ∈ C([0, T], X). + +8 +B. BEKBOLAT AND N. TOKMAGAMBETOV +The classical Mittag-Leffler function Eγ,1(t) and the Mittag-Leffler type function +Eγ,γ(t) are given by the expressions +Eγ,1(t) := +∞ +� +k=0 +tk +Γ(γk + 1) +Eγ,γ(t) := +∞ +� +k=0 +tk +Γ(γk + γ). +In the case γ = 1, we obtain E1,1(t) = et. For more information about the classical +Mittag-Leffler function Eγ,1(t) and the Mittag-Leffler type function Eγ,γ(t) see e.g. +[KST06, p. 40 and p. 42]. +In [Sim14, Theorem 4, p. 21] the following estimate for the Mittag-Leffler function +is proved, when 0 < γ < 1 (not true for γ ≥ 1) +1 +1 + Γ(1 − γ)t ≤ Eγ,1(−t) ≤ +1 +1 + Γ(1 + γ)−1t, +t > 0. +Then it follows +(2.5) +0 < Eγ,1(−t) < 1, +t > 0. +Proposition 2.11. [Pod99] If 0 < γ < 2, β is an arbitrary real number, µ is such +that πγ/2 < µ < min{π, πγ}, then there exists positive constant C, such that we have +|Eγ,β(z)| ≤ +C +1 + |z| +for all µ ≤ | arg(z)| ≤ π. +Lemma 2.12. Assume that 0 < t < T, 0 < γ ≤ 1 and λ ∈ R+. Then +(2.6) +0 < 1 − Eγ,1 (−λtγ) +1 − Eγ,1 (−λT γ) < 1 +and +(2.7) +− 1 < Eγ,1 (−λT γ) − Eγ,1 (−λtγ) +1 − Eγ,1 (−λT γ) +< 0 +inequalities hold. +Proof. Using property (2.5) we have +0 < 1 − Eγ,1 (−λT γ) < 1 +or +(2.8) +1 < +1 +1 − Eγ,1 (−λT γ). +Then multiplying both sides of the inequality (2.8) by 1 − Eγ,1 (−λtγ) we obtain +0 < 1 − Eγ,1 (−λtγ) < 1 − Eγ,1 (−λtγ) +1 − Eγ,1 (−λT γ) < +1 +1 − Eγ,1 (−λT γ) < 1 +and these inequalities imply (2.6). Rewriting the expression +Eγ,1 (−λT γ) − Eγ,1 (−λtγ) +1 − Eγ,1 (−λT γ) += 1 − Eγ,1 (−λtγ) +1 − Eγ,1 (−λT γ) − 1 +and using the first inequality (2.6) we obtain the second inequality (2.7). +□ + +PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR +9 +3. Main Results +In this Section we deal with the direct problem for the time-fractional pseudo- +parabolic equation associated with the Jacobi operator ∆α,β (1.1). Moreover, ISPs +are subject to study. The existence, uniqueness and stability results are established. +3.1. The direct problem for the time-fractional pseudo-parabolic equation +with the Jacobi operator. Let 0 < γ ≤ 1. We consider the non-homogeneous +time-fractional pseudo-parabolic equation +(3.1) Dγ +0+,t (u(t, x) − a∆α,βu(t, x)) − ∆α,βu(t, x) + mu(t, x) = f(t, x), +(t, x) ∈ D, +with initial condition +(3.2) +u(0, x) = φ(x), +x ∈ R+, +where the functions f and φ are given functions. Our aim is to find unique solution +u of the problem (3.1) - (3.2). +Theorem 3.1. Let 0 < γ ≤ 1 and λ ∈ R+. Suppose that f ∈ C1([0, T], L2(µ)) and +φ ∈ H. Then the problem (3.1)-(3.2) has a unique solution u ∈ Cγ([0, T], L2(µ)) ∩ +C([0, T], H) and can be represented by formula +u(t, x) = +� ∞ +0 +� ∞ +0 +� t +0 +(t − τ)γ−1Eγ,γ +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)(t − τ)γ +� +f(τ, y) +1 + a(λ2 + ρ2) +× ϕα,β +λ (y)ϕα,β +λ (x)dτdµα,β(y)dνα,β(λ) ++ +� ∞ +0 +� ∞ +0 +Eγ,1 +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)tγ +� +φ(y)ϕα,β +λ (y)ϕα,β +λ (x)dµα,β(y)dνα,β(λ). +Proof. We assume that 0 < γ ≤ 1, λ ∈ R+ and u(t, ·) ∈ H. We first prove that the +problem (3.1)-(3.2) has only one solution, if the later exists. Suppose the proposi- +tion were false. Assume that there exist two different solutions u1(t, x) and u2(t, x). +Denote u0(t, x) = u1(t, x) − u2(t, x). Then u0(t, x) solves the following equation +(3.3) +Dγ +0+,t (u0(t, x) − a∆α,βu0(t, x)) − ∆α,βu0(t, x) + mu0(t, x) = 0, +(t, x) ∈ D, +(3.4) +u0(0, x) = 0, +x ∈ R+. +The problem (3.3)-(3.4) has only trivial solution. +This implies uniqueness of the +solution. +Now, we will prove the existence of the solutions. Using the Fourier-Jacobi trans- +form Fα,β (2.2) on both sides of (3.1)-(3.2), we have +(3.5) +Dγ +0+,t�u(t, λ) + λ2 + ρ2 + m +1 + a(λ2 + ρ2)�u(t, λ) = +�f(t, λ) +1 + a(λ2 + ρ2), +(3.6) +�u(0, λ) = �φ(λ), +for all λ ∈ R+ and 0 < t < T. The solution (see [KST06, p. 231, ex. 4.9]) of the +problem (3.5)-(3.6) is given by + +10 +B. BEKBOLAT AND N. TOKMAGAMBETOV +(3.7) +�u(t, λ) = +� t +0 +(t − τ)γ−1Eγ,γ +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)(t − τ)γ +� +�f(τ, λ) +1 + a(λ2 + ρ2)dτ ++ �φ(λ)Eγ,1 +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)tγ +� +, +where Eγ,1(z) is the classical Mittag-Leffler function and Eγ,γ(z) is the Mittag-Leffler +type function. Now by using the inverse Fourier-Jacobi transform F −1 +α,β (2.3) to (3.7), +we obtain the formula for the solution of the problem (3.1)-(3.2), given by +u(t, x) = +� ∞ +0 +� ∞ +0 +� t +0 +(t − τ)γ−1Eγ,γ +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)(t − τ)γ +� +f(τ, y) +1 + a(λ2 + ρ2) +× ϕα,β +λ (y)ϕα,β +λ (x)dτdµα,β(y)dνα,β(λ) ++ +� ∞ +0 +� ∞ +0 +Eγ,1 +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)tγ +� +φ(y)ϕα,β +λ (y)ϕα,β +λ (x)dµα,β(y)dνα,β(λ). +By using the property +d +dτ (Eγ,1(cτ γ)) = cτ γ−1Eγ,γ(cτ γ), +c = constant, +of the Mittag-Leffler function, we obtain +d +dτ +� +Eγ,1 +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)(t − τ)γ +�� += +λ2 + ρ2 + m +1 + a(λ2 + ρ2)(t − τ)γ−1Eγ,γ +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)(t − τ)γ +� +and we can write (3.7) in a form +�u(t, λ) = +� t +0 +(t − τ)γ−1Eγ,γ +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)(t − τ)γ +� +�f(τ, λ) +1 + a(λ2 + ρ2)dτ ++ �φ(λ)Eγ,1 +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)tγ +� += +1 +λ2 + ρ2 + m +� t +0 +d +dτ +� +Eγ,1 +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)(t − τ)γ +�� +�f(τ, λ)dτ ++ �φ(λ)Eγ,1 +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)tγ +� += +1 +λ2 + ρ2 + m +�f(t, λ) − +1 +λ2 + ρ2 + m +�f(0, λ)Eγ,1 +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)tγ +� +− +1 +λ2 + ρ2 + m +� t +0 +Eγ,1 +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)(t − τ)γ +� d +dτ +�f(τ, λ)dτ ++ �φ(λ)Eγ,1 +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)tγ +� + +PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR +11 +by using the rule integration by parts and Eγ,1(0) = 1. Let 0 < γ < 1 and f ∈ +C1([0, T], L2(µ)), φ ∈ H, then we can estimate u as follows +∥u(t, ·)∥2 +H = +� ∞ +0 +��(λ2 + ρ2)�u(t, λ) +��2 dνα,β(λ) +≲ +� ∞ +0 +�����(λ2 + ρ2) +�f(t, λ) +λ2 + ρ2 + m +����� +2 +dνα,β(λ) ++ +� ∞ +0 +�����(λ2 + ρ2) +�f(0, λ) +λ2 + ρ2 + mEγ,1 +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)tγ +������ +2 +dνα,β(λ) ++ +� ∞ +0 +���� +λ2 + ρ2 +λ2 + ρ2 + m +� t +0 +Eγ,1 +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)(t − τ)γ +� d +dτ +�f(τ, λ)dτ +���� +2 +× dνα,β(λ) + +� ∞ +0 +����(λ2 + ρ2)�φ(λ)Eγ,1 +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)tγ +����� +2 +dνα,β(λ) +≲ +� ∞ +0 +��� �f(t, λ) +��� +2 +dνα,β(λ) + +� ∞ +0 +��� �f(0, λ) +��� +2 +dνα,β(λ) ++ +� ∞ +0 +�� t +0 +���� +d +dτ +�f(τ, λ) +���� dτ +�2 +dνα,β(λ) + +� ∞ +0 +���(λ2 + ρ2)�φ(λ) +��� +2 +dνα,β(λ) +≲ ∥f(t, ·)∥2 +2,µ + ∥f(0, ·)∥2 +2,µ + +� T +0 +∥ d +dtf(t, ·)∥2 +2,µdt + ∥φ∥2 +H, +here we have used Cauchy-Schwarz inequality, Fubibi’s theorem and a ≲ b denotes +a ≤ cb for some positive constant c independent of a and b. Thus, +∥u(t, ·)∥2 +H ≲ ∥f(t, ·)∥2 +2,µ + ∥f(0, ·)∥2 +2,µ + +� T +0 +∥ d +dtf(t, ·)∥2 +2,µdt + ∥φ∥2 +H. +Then, we obtain +∥u∥2 +C([0,T],H) ≲ ∥f∥2 +C1([0,T],L2(µ)) + ∥φ∥2 +H < ∞. +In a case γ = 1, we have +∥u(t, ·)∥2 +H = +� ∞ +0 +��(λ2 + ρ2)�u(t, λ) +��2 dνα,β(λ) += +� ∞ +0 +�����(λ2 + ρ2) +� t +0 +�f(τ, λ) +1 + a(λ2 + ρ2)e +− λ2+ρ2+m +1+a(λ2+ρ2) (t−τ)dτ ++ (λ2 + ρ2)�φ(λ)e +− λ2+ρ2+m +1+a(λ2+ρ2) t +����� +2 +dνα,β(λ) +≲ +� ∞ +0 +���� +� t +0 +�f(τ, λ)e +− λ2+ρ2+m +1+a(λ2+ρ2) (t−τ)dτ +���� +2 +dνα,β(λ) + +12 +B. BEKBOLAT AND N. TOKMAGAMBETOV ++ +� ∞ +0 +����(λ2 + ρ2)�φ(λ)e +− λ2+ρ2+m +1+a(λ2+ρ2) t +���� +2 +dνα,β(λ) +≲ +� ∞ +0 +� T +0 +��� �f(t, λ) +��� +2 +dtdνα,β(λ) + +� ∞ +0 +���(λ2 + ρ2)�φ(λ) +��� +2 +dνα,β(λ) += +� T +0 +∥f(t, ·)∥2 +2,µdt + ∥φ∥2 +H +by using Cauchy-Schwarz inequality and Fubini’s theorem. Thus, +∥u(t, ·)∥2 +H ≲ +� T +0 +∥f(t, ·)∥2 +2,µdt + ∥φ∥2 +H. +Then, we have +∥u∥2 +C([0,T],H) ≲ ∥f∥2 +C([0,T],L2(µ)) + ∥φ∥2 +H < ∞. +Let us estimate the function Dγ +0+,tu +∥Dγ +0+,tu(t, ·)∥2 +2,µ = ∥Dγ +0+,t�u(t, ·)∥2 +2,ν = +� ∞ +0 +���Dγ +0+,t�u(t, ·) +��� +2 +dνα,β(λ) += +� ∞ +0 +����� +�f(t, λ) +1 + a(λ2 + ρ2) − λ2 + ρ2 + m +1 + a(λ2 + ρ2)�u(t, λ) +����� +2 +dνα,β(λ) +≲ ∥ �f(t, ·)∥2 +2,ν + ∥�u(t, ·)∥2 +2,ν. +Thus, we have +∥Dγ +0+,tu(t, ·)∥2 +2,µ ≲ ∥f(t, ·)∥2 +2,µ + ∥u(t, ·)∥2 +2,µ +and +∥Dγ +0+,tu∥2 +C([0,T],L2(µ)) ≲ ∥f∥2 +C([0,T],L2(µ)) + ∥u∥2 +C([0,T],L2(µ)) < ∞. +Consequently, using Definition 2.10 we obtain u ∈ Cγ([0, T], L2(µ)). Our prove is +completed. +□ +3.2. The ISP for the time-fractional pseudo-parabolic equation. This subsec- +tion deals with the ISP for the time-fractional pseudo-parabolic equation associated +with the Jacobi operator ∆α,β (1.1). +3.2.1. Statement of the problem. Let 0 < γ ≤ 1. We aim to find a couple of functions +(u, f) satisfying equation +(3.8) +Dγ +0+,t (u(t, x) − a∆α,βu(t, x)) − ∆α,βu(t, x) + mu(t, x) = f(x), +(t, x) ∈ D, +under conditions +(3.9) +u(0, x) = φ(x), +x ∈ R+, +and +(3.10) +u(T, x) = ψ(x), +x ∈ R+. + +PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR +13 +Theorem 3.2. Let 0 < γ ≤ 1. Assume that ψ, φ ∈ H. Then the pair (u, f) is a +unique solution of the ISP (3.8)-(3.10), which are functions u ∈ Cγ([0, T], L2(µ)) ∩ +C([0, T], H), f ∈ L2(µ) can be represented by the formulas +u(t, x) = +� ∞ +0 +� ∞ +0 +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)tγ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +�ψ(y)ϕα,β +λ (y)ϕα,β +λ (x)dµα,β(y)dνα,β(λ) +− +� ∞ +0 +� ∞ +0 +Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ� +− Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)tγ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +× φ(y)ϕα,β +λ (y)ϕα,β +λ (x)dµα,β(y)dνα,β(λ) +and +f(x) = +� ∞ +0 +� ∞ +0 +(λ2 + ρ2 + m) +ψ(y) − φ(y)Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +× ϕα,β +λ (y)ϕα,β +λ (x)dµα,β(y)dνα,β(λ). +Proof. We assume that 0 < γ ≤ 1, and u(t, ·), f ∈ H. Let us first prove the existence +result. By using the Fourier-Jacobi transform Fα,β (2.2) on both sides of (3.8)-(3.10), +we obtain +(3.11) +Dγ +0+,t�u(t, λ) + λ2 + ρ2 + m +1 + a(λ2 + ρ2)�u(t, λ) = +�f(λ) +1 + a(λ2 + ρ2), +(t, λ) ∈ D, +(3.12) +�u(0, λ) = �φ(λ), +λ ∈ R+, +(3.13) +�u(T, λ) = �ψ(λ), +λ ∈ R+. +Solution of the equation (3.11) is given by +(3.14) +�u(t, λ) = +�f(λ) +λ2 + ρ2 + m + C(λ)Eγ,1 +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)tγ +� +, +for all 0 < γ ≤ 1 and functions �f(λ) and C(λ) are unknown functions. For determine +these functions we use conditions (3.12) and (3.13). After that we have +�u(0, λ) = +�f(λ) +λ2 + ρ2 + m + C(λ) = �φ(λ) +and +�u(T, λ) = +�f(λ) +λ2 + ρ2 + m + C(λ)Eγ,1 +� +− λ2 + ρ2 + m +1 + a(λ2 + ρ2)T γ +� += �ψ(λ). +Thus we have +C(λ) = +�φ(λ) − �ψ(λ) +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� + +14 +B. BEKBOLAT AND N. TOKMAGAMBETOV +and +(3.15) +�f(λ) = (λ2 + ρ2 + m) +�ψ(λ) − �φ(λ)Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +. +Substituting the resulting functions C(λ) and �f(λ) into (3.14), we get +�u(t, λ) = +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)tγ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� �ψ(λ) +− +Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ� +− Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)tγ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +�φ(λ). +Therefore solution of the problem (3.11) - (3.13) is the pair (�u, �f). We obtain solution +of the problem (3.8)-(3.10) by applying the inverse Fourier-Jacobi transform F −1 +α,β (2.3) +to the functions �u and �f, i.e. +(3.16) +u(t, x) = +� ∞ +0 +� ∞ +0 +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)tγ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +�ψ(y)ϕα,β +λ (y)ϕα,β +λ (x)dµα,β(y)dνα,β(λ) +− +� ∞ +0 +� ∞ +0 +Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ� +− Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)tγ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +× φ(y)ϕα,β +λ (y)ϕα,β +λ (x)dµα,β(y)dνα,β(λ) +and +(3.17) +f(x) = +� ∞ +0 +� ∞ +0 +(λ2 + ρ2 + m) +ψ(y) − φ(y)Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +× ϕα,β +λ (y)ϕα,β +λ (x)dµα,β(y)dνα,β(λ), +for all 0 < γ ≤ 1. +Let ψ, φ ∈ H. Then using Lemma 2.12 we can estimate the function u as following +∥u(t, ·)∥2 +H = +� ∞ +0 +|(λ2 + ρ2)�u(t, λ)|2dνα,β(λ) +≲ +� ∞ +0 +������ +(λ2 + ρ2) �ψ(λ) +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)tγ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +������ +2 +dνα,β(λ) + +PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR +15 ++ +� ∞ +0 +������ +(λ2 + ρ2)�φ(λ) +Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ� +− Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)tγ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +������ +2 +dνα,β(λ) +≲ +� ∞ +0 +|(λ2 + ρ2) �ψ(λ)|2dνα,β(λ) + +� ∞ +0 +|(λ2 + ρ2)�φ(λ)|2dνα,β(λ). +Thus, +∥u(t, ·)∥2 +H ≲ ∥ψ∥2 +H + ∥φ∥2 +H < ∞. +Then we have +∥u∥C([0,T],H) ≲ ∥ψ∥H + ∥φ∥H < ∞. +Let us estimate the function f +∥f∥2 +2,µ = ∥ �f∥2 +2,ν = +� ∞ +0 +| �f(λ)|2dνα,β(λ) += +� ∞ +0 +������ +(λ2 + ρ2 + m) +�ψ(λ) − �φ(λ)Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +������ +2 +dνα,β(λ) +≲ +� ∞ +0 +������ +(λ2 + ρ2 + m) +�ψ(λ) +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +������ +2 +dνα,β(λ) ++ +� ∞ +0 +������ +(λ2 + ρ2 + m) +�φ(λ)Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +������ +2 +dνα,β(λ) +≲ ∥ψ∥2 +H + ∥φ∥2 +H. +So, we obtain +∥f∥2 +2,µ ≲ ∥ψ∥2 +H + ∥φ∥2 +H < ∞. +Next, we estimate the function Dγ +0+,tu +∥Dγ +0+,tu(t, ·)∥2 +2,µ = ∥Dγ +0+,t�u(t, ·)∥2 +2,ν = +� ∞ +0 +|Dγ +0+,t�u(t, λ)|2dνα,β(λ) += +� ∞ +0 +����� +�f(λ) +1 + a(λ2 + ρ2) − λ2 + ρ2 + m +1 + a(λ2 + ρ2)�u(t, λ) +����� +2 +dνα,β(λ) +≲ ∥f∥2 +2,ν + ∥u(t, ·)∥2 +2,ν. +Finally, we have +∥Dγ +0+,tu∥2 +C([0,T],L2(µ)) ≲ ∥f∥2 +2,µ + ∥u∥2 +C([0,T],L2(µ)) < ∞. +It is obvious that ∥u∥2 +C([0,T],L2(µ)) < ∞. The existence is proved. +Now, let us prove the uniqueness of the solution. Taking into account the property +of the Fourier-Jacobi transform Proposition 2.4, one observes that a pair of functions +(u, f) is uniquely determined by the formulas (3.16) and (3.17). The uniqueness is +proved. +□ + +16 +B. BEKBOLAT AND N. TOKMAGAMBETOV +3.2.2. Stability Theorem. Finally, we study a stability property of the solution (u, f) +of the problem (3.8)-(3.10) given by the formulas (3.16) and (3.17), . +Theorem 3.3. Let (u, f) and (ud, fd) be solutions of the problem (3.8)-(3.10) corre- +sponding to the data (φ, ψ) and its small perturbation (φd, ψd), respectively. Then the +solution of the problem (3.8)-(3.10) depends continuously on these data, namely, we +have +∥u − ud∥2 +C([0,T],H) ≲ ∥ψ − ψd∥2 +H + ∥φ − φd∥2 +H +and +∥f − fd∥2 +2,µ ≲ ∥ψ − ψd∥2 +H + ∥φ − φd∥2 +H. +Proof. From the definition of the Fourier-Jacobi transform (2.2) +Fα,β(u(t, ·))(λ) = �u(t, λ) = +� ∞ +0 +u(t, x)ϕα,β +λ (x)dµα,β(x), +we conclude that +Fα,β(u(t, ·) − ud(t, ·))(λ) = +� ∞ +0 +(u(t, x) − ud(t, x))ϕα,β +λ (x)dµα,β(x) += +� ∞ +0 +u(t, x)ϕα,β +λ (x)dµα,β(x) +− +� ∞ +0 +ud(t, x)ϕα,β +λ (x)dµα,β(x) += Fα,β(u(t, ·))(λ) − Fα,β(ud(t, ·))(λ) += �u(t, λ) − �ud(t, λ), +here we have used property of the integral. According to the above statement and +using Lemma 2.12, we have +∥u(t, ·) − ud(t, ·)∥2 +H = +� ∞ +0 +(λ2 + ρ2)2|�u(t, λ) − �ud(t, λ)|2dνα,β(λ) += +� ∞ +0 +(λ2 + ρ2)2 +����� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)tγ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +� +�ψ(λ) − �ψd(λ) +� +− +Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ� +− Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)tγ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +� +�φ(λ) − �φd(λ) +� ���� +2 +dνα,β(λ) +≲ +� ∞ +0 +(λ2 + ρ2)2 ��� �ψ(λ) − �ψd(λ) +��� +2 +dνα,β(λ) + +� ∞ +0 +(λ2 + ρ2)2 ����φ(λ) − �φd(λ) +��� +2 +dνα,β(λ) += ∥ψ − ψd∥2 +H + ∥φ − φd∥2 +H. +Thus, one gets +∥u(t, ·) − ud(t, ·)∥2 +H ≲ ∥ψ − ψd∥2 +H + ∥φ − φd∥2 +H, +and +∥u − ud∥2 +C([0,T],H) ≲ ∥ψ − ψd∥2 +H + ∥φ − φd∥2 +H. + +PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR +17 +By writing (3.15) in the form +�f(λ) = +λ2 + ρ2 + m +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� �ψ(λ) − +(λ2 + ρ2 + m)Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +�φ(λ), +and applying similar estimates again we can observe that +∥f − fd∥2 +2,µ = ∥ �f − �fd∥2 +2,ν = +� ∞ +0 +��� �f(λ) − �fd(λ) +��� +2 +dνα,β(λ) += +� ∞ +0 +����� +λ2 + ρ2 + m +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +� +�ψ(λ) − �ψd(λ) +� +− +(λ2 + ρ2 + m)Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +� +�φ(λ) − �φd(λ) +������ +2 +dνα,β(λ) +≲ +� ∞ +0 +������ +λ2 + ρ2 + m +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +� +�ψ(λ) − �ψd(λ) +� +������ +2 +dνα,β(λ) ++ +� ∞ +0 +������ +(λ2 + ρ2 + m)Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ� +1 − Eγ,1 +� +− λ2+ρ2+m +1+a(λ2+ρ2)T γ +� +� +�φ(λ) − �φd(λ) +� +������ +2 +dνα,β(λ) +≲ +� ∞ +0 +(λ2 + ρ2)2 ��� �ψ(λ) − �ψd(λ) +��� +2 +dνα,β(λ) + +� ∞ +0 +(λ2 + ρ2)2 ����φ(λ) − �φd(λ) +��� +2 +dνα,β(λ) += ∥ψ − ψd∥2 +H + ∥φ − φd∥2 +H. +It follows easily that +∥f − fd∥2 +2,µ ≲ ∥ψ − ψd∥2 +H + ∥φ − φd∥2 +H, +ending the proof. +□ +3.2.3. Stability Test. Here to check Theorem 3.3 we consider a ISP for the heat equa- +tion with one dimensional Sturm-Liouville operator +(3.18) +ut(t, x) − uxx(t, x) = f(x), +0 < t < 1, +x > 0, +with conditions +(3.19) +u(0, x) = u(1, x) = 0, +where we put T = γ = 1, α = β = −1 +2, a = m = 0 and φ(x) = ψ(x) = 0 for all x > 0. +Also, consider a perturbed problem with some noise +uǫ +t(t, x) − uǫ +xx(t, x) = f ǫ(x), +0 < t < 1, +x > 0, +with conditions +uǫ(0, x) = 0, +and +uǫ(1, x) = ǫ · e−x2, +x > 0, + +18 +B. BEKBOLAT AND N. TOKMAGAMBETOV +and with additional information φǫ(x) = 0 and ψǫ(x) = ǫ · e−x2, where ǫ is a positive +constant. Then by Theorem 3.2, we have +uǫ(t, x) = +ǫ +√π +� ∞ +0 +1 − e−λ2t +1 − e−λ2 e− λ2 +4 cos(λx)dλ, +and +f ǫ(x) = +ǫ +√π +� ∞ +0 +λ2e− λ2 +4 +(1 − e−λ2) cos(λx)dλ. +Illustrations of our calculations above are given in Table 1. +ǫ +1 +0.2 +0.02 +∥ψ − ψǫ∥2 +H +1.5 +0.06 +0.0006 +∥u − uǫ∥2 +C([0,1],H) +0.75 +0.03 +0.0003 +∥f − f ǫ∥2 +2,µ +1.0474 +0.041897 +0.0004 +Table 1. Stability Test +Conclusion. +Table 1 confirms that the solution of the problem (3.8)-(3.10) is +continuously depending on the given data. Small changes in the given data imply +small changes in (u, f). +4. Appendix +Calculations in Table 1 are made by using Maple 2021 program with the following +codes: +psi := exp(−x2), +hat(psi) := int +� +1 +√ +2π +· psi · cos(x · λ), x = 0..∞ +� +, +norm(psi) := 40 · int +� +4 +√ +2π +· λ4 · |hat(psi)|2, λ = 0..∞ +� +, +hat(u) := 1 − exp(−λ2 · t) +1 − exp(−λ2) +· hat(psi), +norm(u) := int +� +4 +√ +2π +· λ4 · |hat(u)|2, λ = 0..∞ +� +, +lim +t→1(norm(u)), +hat(f) := λ2 · hat(psi) +1 − exp(−λ2), +and +norm(f) := int +� +4 +√ +2π +· |hat(f)|2, λ = 0..∞ +� +. + +PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR +19 +References +[Bus95] I. Bushuyev, Global uniqueness for inverse parabolic problems with final observation, In- +verse Problems, 11 (1995), L11-L16. +[CF18] P.M. de Carvalho-Neto, R. Fehlberg J´unior, Conditions for the absence of blowing up solu- +tions to fractional differential equations, Acta Appl Math, 154 (2018), 15-29. +[CNYY09] J. Cheng, J. Nakagawa, M. Yamamoto, T. 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Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse +Problems, 31 (2015), 035003. +[Koo75] T.H. Koorwinder, A new proof of a Paley-Wiener type theorem for the Jacobi transform, +Ark. Mat., 13 (1975), 145-159. +[KS10] I.A. Kaliev, M.M. Sabitova, Problems of determining the temperature and density of heat +sources from the initial and final temperatures, Journal of Applied and Industrial Mathematics, +4:3 (2010), 332-339. +[KST17] M. Kirane, B. Samet, B.T. Torebek, Determination of an unknown source term tempera- +ture distribution for the sub-diffusion equation at the initial and final data, Electronic Journal +of Differential Equations 2017:257 (2017), 1–13. +[KST06] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differ- +ential Equations, Elsevier, North-Holland, Mathematics studies, 2006. +[OS12a] I. Orazov, M.A. Sadybekov, One nonlocal problem of determination of the temperature +and density of heat sources, Russian Mathematics, 56:2 (2012), 60-64. +[OS12b] I. Orazov, M.A. Sadybekov, On a class of problems of determining the temperature and +density of heat sources given initial and final temperature, Siberian Mathematical Journal, 53:1 +(2012), 146-151. +[Pod99] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. +[PT92] A. I. Prilepko, I. V. Tikhonov, Uniqueness of a solution of the inverse problem for the +evolution equation and application to the transport equation, Mathematical Notes, 51 (1992), +158–165. +[Run80] W. Rundell, Determination of an unknown nonhomogeneous term in a linear partial dif- +ferential equation from overspecified boundary data. Appl. Anal., 10 (1980), 231–242. +[RSTT21] M. Ruzhansky, D. Serikbaev, B.T. Torebek, N. Tokmagambetov, Direct and inverse +problems for time-fractional pseudo-parabolic equations, Quaestiones Mathematicae, 2021, DOI: +10.2989/16073606.2021.1928321. +[RTT19] M. Ruzhansky, N. Tokmagambetov, B.T. Torebek, Inverse source problems for positive +operators. I: Hypoelliptic diffusion and subdiffusion equations, Journal of Inverse and Ill-Posed +Problems, 2019. +[SY11] K. Sakamoto and M. Yamamoto, Inverse source problem with a final overdetermination for +a fractional diffusion equation, Mathematical control and related fields, 1:4 (2011), 509-518. +[SD98] N.B. Salem, A. Dachraoui, Pseudo-differential operators associated with the Jacobi differ- +ential operator, J. Math. Anal. Appl., 220 (1998), 365-381. +[SD00] N.B. Salem, A. Dachraoui, Sobolev type spaces associated with Jacobi differential operators, +Integral Transforms and Special Functions, 9 (2000), 163-184. + +20 +B. BEKBOLAT AND N. TOKMAGAMBETOV +[SS11] N.B. Salem, T. Samaali, Hilbert transform and related topics associated with the differential +Jacobi operator on (0, +∞), Positivity, 15 (2011), 221-240. +[Sim14] T. Simon, Comparing Frechet and positive stable laws, Electron. J. Probab., 91 (2014), +1-25. +[Slo13] M. Slodi˘cka, A source identification problem in linear parabolic problems: A semigroup +approach, Journal of Inverse and Ill-Posed Problems, 21 (2013), 579-600. +[SS16] M. Slodi˘cka, M. ˘Si˘skova, An inverse source problem in a semilinear time-fractional diffusion +equation, Computers and Mathematics with Applications, 72 (2016), 1655–1669. +[SSB19] M.M. Slodi˘cka, M. ˘Si˘skova, K. V. Bockstal, Uniqueness for an inverse source problem of +determining a space dependent source in a time-fractional diffusion equation, Appl. Math. Lett., +91 (2019), 15–21. +[TE02] I.V. Tikhonov and Yu.S. Eidelman, An inverse problem for a differential equation in a Ba- +nach space and distribution of zeros of an entire Mittag-Leffler function, Differential Equations, +38:5 (2002), 669-677. +[TT17] B.T. Torebek, R. Tapdigoglu, Some inverse problems for the nonlocal heat equation with +Caputo fractional derivative, Mathematical Methods in the Applied Sciences, 40:18 (2017), +6468–6479. +[WYH13] W. Wang, M. Yamamoto, B. Han, Numerical method in reproducing kernel space for +an inverse source problem for the fractional diffusion equation, Inverse Problems, 29 (2013), +095009. +[YG03] M. Yaman, O.F. G¨oz¨ukızıl, Asymptotic behaviour of the solutions of inverse problems for +pseudo-parabolic equations, Applied Mathematics and Computation, 154 (2004), 69–74. +[Yam12] M. Yaman, Blow-up solution and stability to an inverse problem for a pseudo-parabolic +equation, Journal of Inequalities and Applications, 2012 (2012),274. +Bayan Bekbolat : +Al-Farabi Kazakh National University +Almaty, Kazakhstan +and +Department of Mathematics: Analysis, Logic and Discrete Mathematics +Ghent University, Belgium +and +Institute of Mathematics and Mathematical Modeling +Almaty, Kazakhstan +and +Suleyman Demirel University +Kaskelen, Kazakhstan +E-mail address: bekbolat@math.kz +Niyaz Tokmagambetov: +Centre de Recerca Matem´atica +Campus de Bellaterra, Edifici C, 08193 Barcelona, Spain +and +Institute of Mathematics and Mathematical Modeling +Almaty, Kazakhstan +E-mail address tokmagambetov@crm.cat; tokmagambetov@math.kz + diff --git a/MdAyT4oBgHgl3EQfs_ko/content/tmp_files/load_file.txt b/MdAyT4oBgHgl3EQfs_ko/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fa8f5e1f6c1f0aedd05a7e96585e4f060c47248d --- /dev/null +++ b/MdAyT4oBgHgl3EQfs_ko/content/tmp_files/load_file.txt @@ -0,0 +1,760 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf,len=759 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='00585v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='AP] 2 Jan 2023 INVERSE SOURCE PROBLEM FOR THE PSEUDO-PARABOLIC EQUATION ASSOCIATED WITH THE JACOBI OPERATOR BAYAN BEKBOLAT AND NIYAZ TOKMAGAMBETOV Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' In this paper we investigate direct and inverse problems for time- fractional pseudo-parabolic equations associated with the Jacobi operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The existence and uniqueness of the solutions are proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Also, the stability result of the inverse source problem (ISP) is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Introduction The main object of this paper is the following non-homogeneous time-fractional pseudo-parabolic equation on the domain D = {(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' x) : 0 < t < T < ∞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' x ∈ R+ = (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ∞)} Dγ 0+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='t (u(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' x) − a∆α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='βu(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' x)) − ∆α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='βu(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' x) + mu(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' x) = f(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' where 0 < γ ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' with non–negative constants m and a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' and with the initial condition u(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' x) = φ(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' x ∈ R+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' where Dγ 0+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='t is given by Dγ 0+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='t = � Dγ 0+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 0 < γ < 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' d dt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' γ = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Dγ 0+,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='t is the left-sided Caputo fractional derivative and ∆α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β is the Jacobi operator given by the expression (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1) ∆α,β = A−1 α,β(x) d dx � Aα,β(x) d dx � , x ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Here, we denote by Aα,β(x) = 22ρ(sinh(x))2α+1(cosh(x))2β+1, ρ = α + β + 1, with α ≥ β ≥ −1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' In our studies we would be questioned about the well–posedness of the direct prob- lem and the stability of the inverse source problem with the additional information – over-determination condition u(T, x) = ψ(x), x ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' For the ISP we will restore the pair (u, f) under some conditions on the function ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Date: January 3, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Primary 35R30;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Secondary 35R11, 35C15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Jacobi operator, Jacobi transform, time-fractional pseudo-parabolic equation, inverse source problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' This research was funded by the Science Committee of the Ministry of Science and Higher Edu- cation of the Republic of Kazakhstan (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' AP14972634).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 1 2 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' BEKBOLAT AND N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' TOKMAGAMBETOV One of the first mathematicians who studied the ISP was Rundell [Run80] in 1980s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' He considered the evolution type equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2) du dt + Au = f in a Banach space X, where A is linear operator in X and f is a constant vector in X, with conditions u(0) = u0, and u(T) = u1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Using semigroups of operators Rundell proved a general theorem about the existence of a unique solution pair (u(t), f) of the problem, which then was applied to equa- tions of parabolic and pseudo-parabolic types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' When the non-homogeneous term is represented in the form f(t) = Φ(t)f, where Φ(t) is known operator and the element f is unknown, and A is a closed linear operator from Lp(Ω) into Lp(Ω), several ISPs for the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2) were studied by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Prilepko and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Tikhonov [PT92] in 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' They applied obtained results to the transport equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' In the general case, where the unknown source depends on time, under a sufficient condition, ISPs for the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2) with the linear elliptic partial differential operator A of order 2m with the bounded measurable coefficients such that (Aϕ, ϕ) ≥ ∥ϕ∥2 for all ϕ ∈ H2m(Ω) ∩ Hm 0 (Ω), µ = constant > 0 was investigated by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Bushuyev [Bus95] in 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Nonetheless, there is no general closed theory for abstract case of F(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Known results deal with separated source terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' In 2002 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Tikhonov and Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Eidelman [TE02] considered ISPs for the generalization of the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2) of the form dNu(t) dtN = Au(t) + p, 0 < t < T, for some positive integer N ≥ 1 and some real number T > 0 with an unknown parameter p and a closed linear operator A in the Banach space under the Cauchy conditions and ”over-determination condition” u(T) = uN (also in the Banach space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' For the Laplace operator (−∆) which is one of the most interesting examples in Physics, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Choulli and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Yamamoto in [CY04] established the uniqueness and conditional stability in determining a heat source term from boundary measurements with f = σ(t)ϕ(x), where σ(t) is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Asymptotic behaviour of the solution of the inverse source problem for the pseudo- parabolic equation (u(x, t) − ∆u(x, t))t − ∆u(x, t) + αu(x, t) = f(t)g(x, t), Q∞ = Ω × (0, ∞) with a integral over-determination condition was studied by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Yaman and ¨O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' G¨oz¨ukızıl in [YG03] in 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Fractional derivatives and fractional partial differential equations have received great attention both in analysis and application, which are used in modeling several phenomena in different areas of science such as biology, physics, and chemistry, so the fractional computation is increasingly attracted to mathematicians in the last several decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ISP for the time fractional parabolic equation cDα t u(x, t) = rα(Lu)(x, t) + f(x)h(x, t), x ∈ Ω, t ∈ (0, T), 0 < α < 1, PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR 3 where cDα t is the Caputo derivative defined by cDα t g(t) = 1 Γ(1 − α) � t 0 (t − τ)−α d dτ g(τ)dτ and L is a symmetric uniformly elliptic operator was considered by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Sakamoto and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Yamamoto in [SY11] in 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The authors proved that the inverse problem is well-posed in the Hadamard sense except for a discrete set of values of diffusion constants using final overdetermining data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Blow-up solution and stability to ISP for the pseudo-parabolic equation ut − a∆ut − ∆u + n � i=1 biuxi − |u|pu = f(t)g(t), x ∈ Ω, t > 0 with the integral overdetermination condition was studied by Metin Yaman in [Yam12] in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ISP for the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2) considered by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Slodi˘cka in [Slo13] in 2013, when A is a linear differential operator of second-order, strongly elliptic, and the right- hand side f is assumed to be separable in both variables x and t, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' f(x, t) = g(x)h(t) (in this case h(t) is unknown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ISP for a semilinear time-fractional diffusion equation of second order in a bounded domain in Rd (g1−β ∗ ∂tu(x))(t) + L(x, t)u(x, t) = h(t)f(x) + � t 0 F(x, s, u(x, s))ds with a linear second order differential operator L(x, t) in the divergence form with space and time dependent coefficients was studied by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Slodi˘cka and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ˘Si˘skova in [SS16] in 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Authors showed the existence, uniqueness and regularity of a weak solution (u, h) ([SS16, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 1658]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' One of the recent papers for inverse source problems for pseudo-parabolic equations with fractional derivatives is [RSTT21] (in 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' In [RSTT21], authors have considered solvability of an in- verse source problem for the pseudo-parabolic equation with the Caputo fractional derivative Dα t of order 0 < α ≤ 1 Dα t (u(t) + Lu(t)) + Mu(t) = f(t) in H, u(0) = φ ∈ H, u(T) = ψ ∈ H, where H be a separable Hilbert space and L, M be operators with the corresponding discrete spectra on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The authors obtained well-posedness results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' A number of articles address the solvability of the inverse problems for the diffusion and sub-diffusion equations ([CNYY09, JR15, KS10, KST17, OS12a, OS12b, RTT19]) and fractional diffusion equations ([SSB19, TT17, WYH13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The semigroups (H(α,β) t )t≥0 (the solution of the heat equation associated with the Jacobi-Dunkl operator Λ2 α,β ) generate a new family of Markov processes on the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' On some Riemannian symmetric spaces this process is the radial part of the Brownian motion for particular values of (α, β) [CGM06].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' However, the ISP for the pseudo-parabolic equations generated by the Jacobi op- erator ∆α,β (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1) have not been still considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' So, our goal is to consider the ISP for the pseudo-parabolic equation with this special operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Harmonic analysis associated with the operator ∆α,β has been studied by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Flensted-Jensen and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Koornwinder [FJ72, FJK73, FJK79, Koo75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The spectral decomposition of the 4 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' BEKBOLAT AND N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' TOKMAGAMBETOV Jacobi operator was considered by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Flensted-Jensen in 1972 [FJ72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' There were obtained a generalization of the classical Paley-Wiener Theorem and a generalized Fourier transform Fα,β, is called Jacobi-Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Eigenfunctions ϕα,β λ (x) of the operator Jacobi is called the Jacobi function, which is hypergeometric func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The pseudo-differential operators (see [SD98]) and Sobolev type spaces Gs,p α,β (see [SD00]) associated with the Jacobi operator was studied by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Ben Salem and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Dachraoui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' In [SD98], authors proved that a pseudo-differential operator associ- ated with a symbol in Sm 0 is a continuous linear mapping from some subspace of the Schwartz space into itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Our main result reads as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Let 0 < γ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Assume that ψ, φ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Then the pair (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' f) is a unique solution of the ISP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' which are functions u ∈ Cγ([0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' T],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' L2(µ))∩C([0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' T],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' H),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' f ∈ L2(µ) can be represented by the formulas u(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' x) = � ∞ 0 � ∞ 0 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)tγ� 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ �ψ(y)ϕα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β λ (y)ϕα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β λ (x)dµα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(y)dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) − � ∞ 0 � ∞ 0 Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ� − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)tγ� 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � × φ(y)ϕα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β λ (y)ϕα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β λ (x)dµα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(y)dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) and f(x) = � ∞ 0 � ∞ 0 (λ2 + ρ2 + m) ψ(y) − φ(y)Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ� 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � × ϕα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β λ (y)ϕα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β λ (x)dµα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(y)dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The contents of this paper as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' In Section 2, we collect some results about harmonic analysis associated with the Jacobi operator on R+ and here we introduce the Sobolev type space H, also given some necessary information about fractional derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' In Section 3, we prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 for the direct problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' In Section 4, we prove our main Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2 about solvability of the inverse source problem associated with the Jacobi operator on R+, also shown stability analysis and example for the inverse source problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Jacobi analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The singular second order differential equation ([FJ72]) ∆α,βϕα,β λ (x) + (λ2 + ρ2)ϕα,β λ (x) = 0 on (0, ∞) with initial conditions ϕα,β λ (0) = 1, d dtϕα,β λ (0) = 0 PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR 5 has a unique solution, given by the expression (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1) ϕα,β λ (x) = 2F1 �1 2(ρ + iλ), 1 2(ρ − iλ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' α + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' − sinh2 x � , where 2F1 is the Gauss hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The function ϕα,β λ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1) is called the Jacobi function and analytic for x ∈ [0, ∞) and ϕα,β λ (x) = ϕα,β −λ (x) and ϕα,β λ (x) = ϕα,β λ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' In particularly, we have ϕ − 1 2,− 1 2 λ (x) = cos(λx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ([FJ72, Proposition 1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 144]) For each fixed x ∈ (0, ∞), as a function of λ, ϕα,β λ is an entire function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Properties of the Jacobi function: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' For all λ ∈ C and x ∈ [0, ∞), we have ([FJ72, Lemma 11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 153]) i) |ϕα,β λ (x)| ≤ ϕα,β iImλ(x), ii) If |Imλ| ≥ ρ then |ϕα,β λ (x)| ≤ e(|Imλ|−ρ)x, iii) If |Imλ| ≤ ρ then |ϕα,β λ (x)| ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' For all n ∈ Z+ there exists Kn > 0 such that ([FJ72, Theorem 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 145]) ���� dn dxnϕα,β λ (x) ���� ≤ Kn(1 + x)(1 + |λ|)ne(|Imλ|−ρ)x and ���� dn dλnϕα,β λ (x) ���� ≤ Kn(1 + x)n+1e(|Imλ|−ρ)x for all λ ∈ C, x ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Let us introduce the following functions spaces ([FJ72, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 146-147], [SD98, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 368]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Let Se(R) be the space of even, rapidly decreasing, and C∞-functions on R, equipped with usual Schwartz topology, and Sr e(R) = {(cosh x) −2ρ r Se(R)}, 0 < r ≤ 2 be the space with the topology defined by the semi-norms Nn,k(f) = sup x≥0 (cosh x) 2ρ r (1 + x)n ���� dk dxk f(x) ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Clearly Sr e(R) is invariant under ∆α,β and the semi-norms defined by Nn,k(f) = sup x≥0 (cosh x) 2ρ r (1 + x)n|∆k α,βf(x)| are continuous on Sr e(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Let Lp(R+, µα,β), 1 ≤ p < ∞ be the space of measurable functions f on R+ such that ∥f∥p p,µ = � ∞ 0 |f(x)|pdµα,β(x) < ∞, where dµα,β(x) = (2π)− 1 222ρ(sinh x)2α+1(cosh x)2β+1dx or dµα,β(x) = (2π)− 1 2Aα,β(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' [FJ72, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 146] Notice that Sr e(R) ⊂ Lr(R+, µα,β) for all 0 < r ≤ 2 and if r ≤ s then Sr e(R) ⊆ Ss e(R) ⊂ L2(R+, µα,β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' BEKBOLAT AND N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' TOKMAGAMBETOV Let Lp(R+, να,β), 1 ≤ p < ∞ be the space of measurable functions g on R+ such that ∥f∥p p,ν = � ∞ 0 |g(λ)|pdνα,β(λ) < ∞, where dνα,β(λ) = (2π)− 1 2|cα,β(λ)|−2dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Here, cα,β(λ) is the Harish–Chandra function, given by cα,β(λ) = 2ρ−iλΓ(iλ)Γ(α + 1) Γ( ρ+iλ 2 )Γ( α−β+1+iλ 2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' For short, we use notations Lp(µ) and Lp(ν) instead Lp(R+, µα,β) and Lp(R+, να,β), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' For f ∈ L1(µ) the Fourier-Jacobi transform Fα,β of f is defined by ([FJ72, Propo- sition 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 146], [SD98, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 369]) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2) �f(λ) = (Fα,βf)(λ) = � ∞ 0 f(x)ϕα,β λ (x)dµα,β(x) and for g ∈ L1(ν) the inverse Fourier-Jacobi transform F −1 α,β is given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='3) � F −1 α,βg � (x) = � ∞ 0 g(λ)ϕα,β λ (x)dνα,β(λ), where ϕα,β λ is the Jacobi functions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ([FJ72, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 145-146]) The operator in L2(µ) defined by ∆α,β with domain D0 α,β = {u ∈ L2(µ) : u and u′ are absolutely continuous and ∆α,βu ∈ L2(µ)} can be restricted to a domain Dα,β, such that ∆α,β becomes self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ∆α,β contains at least functions in D0 α,β which are differentiable at zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ∆α,β has limit-point at ∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' and at zero there is limit-point if 2α + 1 ≥ 3, and limit-circle if 2α + 1 < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' In this last case Dα,β ̸= D0 α,β and choosing λ1 ∈ C with Imλ2 1 > 0 we can define Dα,β = {u ∈ D0 α,β : lim x→0(Aα,β(x) · (ϕα,β λ1 (x)u′(x) − � d dxϕα,β λ1 (x) � u(x))) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ([FJ72, Proposition 3, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 146]) For f ∈ L2(µ) and λ ∈ R+ define �f the integral converging in L2(ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' f → �f is a linear, normpreserving map of L2(µ) onto L2(ν), the inverse given by f(x) = � ∞ 0 g(λ)ϕα,β λ (x)dνα,β(λ) the integral converging in L2(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' A function f ∈ L2(µ) belongs to Dα,β if and only if (λ2 + ρ2) �f(λ) ∈ L2(ν) and in that case � ∆α,βf(λ) = −(λ2 + ρ2) �f(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' In particularly, we have for Plancherel’s identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='4) ∥ �f∥2,ν = ∥f∥2,µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR 7 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' For α = β = −1 2, we have the Fourier-cosine transform �fc(λ) = (Fcf)(λ) = 1 √ 2π � ∞ 0 cos(λx)f(x)dx, and the inverse Fourier-cosine transform is defined by � F −1 c g � (x) = 4 √ 2π � ∞ 0 cos(λx)g(λ)dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' We define the space H := {u ∈ L2(µ) : (·2 + ρ2)�u ∈ L2(ν)} with norm ∥u∥2 H := � ∞ 0 |(λ2 + ρ2)�u(λ)|2dνα,β(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Fractional differentiation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' In this subsection, we introduce frac- tional differentiation operators and other conceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' [KST06, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 69] Let [a, b] (−∞ < a < b < ∞) be a finite interval on the real axis R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The left and right Riemann-Liouville fractional integrals Iγ a+ and Iγ b− of order γ ∈ R (γ > 0) are defined by Iγ a+[f](t) := 1 Γ(γ) � t a (t − s)γ−1f(s)ds, t ∈ (a, b], and Iγ b−[f](t) := 1 Γ(γ) � b t (t − s)γ−1f(s)ds, t ∈ [a, b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Here Γ denotes the Euler gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' [KST06, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 70] The left and right Riemann-Liouville fractional derivatives Dγ a+ and Dγ b− of order γ ∈ R (0 < γ < 1) are given by Dγ a+[f](t) := d dtI1−γ a+ [f](t), ∀t ∈ (a, b], and Dγ b−[f](t) := − d dtI1−γ b− [f](t), ∀t ∈ [a, b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' [KST06, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 91] The left and right Caputo fractional derivatives Dγ a+ and Dγ b− of order γ ∈ R (0 < γ < 1) are defined by Dγ a+[f](t) := Dγ a+[f(t) − f(a)], t ∈ (a, b], and Dγ b−[f](t) := Dγ b−[f(t) − f(b)], t ∈ [a, b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' [CF18, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 18, Definition 3] Let X be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' We say that u ∈ Cγ([0, T], X) if u ∈ C([0, T], X) and Dγ t u ∈ C([0, T], X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 8 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' BEKBOLAT AND N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' TOKMAGAMBETOV The classical Mittag-Leffler function Eγ,1(t) and the Mittag-Leffler type function Eγ,γ(t) are given by the expressions Eγ,1(t) := ∞ � k=0 tk Γ(γk + 1) Eγ,γ(t) := ∞ � k=0 tk Γ(γk + γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' In the case γ = 1, we obtain E1,1(t) = et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' For more information about the classical Mittag-Leffler function Eγ,1(t) and the Mittag-Leffler type function Eγ,γ(t) see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' [KST06, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 40 and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' In [Sim14, Theorem 4, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 21] the following estimate for the Mittag-Leffler function is proved, when 0 < γ < 1 (not true for γ ≥ 1) 1 1 + Γ(1 − γ)t ≤ Eγ,1(−t) ≤ 1 1 + Γ(1 + γ)−1t, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Then it follows (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='5) 0 < Eγ,1(−t) < 1, t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' [Pod99] If 0 < γ < 2, β is an arbitrary real number, µ is such that πγ/2 < µ < min{π, πγ}, then there exists positive constant C, such that we have |Eγ,β(z)| ≤ C 1 + |z| for all µ ≤ | arg(z)| ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Assume that 0 < t < T, 0 < γ ≤ 1 and λ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='6) 0 < 1 − Eγ,1 (−λtγ) 1 − Eγ,1 (−λT γ) < 1 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='7) − 1 < Eγ,1 (−λT γ) − Eγ,1 (−λtγ) 1 − Eγ,1 (−λT γ) < 0 inequalities hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Using property (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='5) we have 0 < 1 − Eγ,1 (−λT γ) < 1 or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='8) 1 < 1 1 − Eγ,1 (−λT γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Then multiplying both sides of the inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='8) by 1 − Eγ,1 (−λtγ) we obtain 0 < 1 − Eγ,1 (−λtγ) < 1 − Eγ,1 (−λtγ) 1 − Eγ,1 (−λT γ) < 1 1 − Eγ,1 (−λT γ) < 1 and these inequalities imply (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Rewriting the expression Eγ,1 (−λT γ) − Eγ,1 (−λtγ) 1 − Eγ,1 (−λT γ) = 1 − Eγ,1 (−λtγ) 1 − Eγ,1 (−λT γ) − 1 and using the first inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='6) we obtain the second inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' □ PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Main Results In this Section we deal with the direct problem for the time-fractional pseudo- parabolic equation associated with the Jacobi operator ∆α,β (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Moreover, ISPs are subject to study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The existence, uniqueness and stability results are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The direct problem for the time-fractional pseudo-parabolic equation with the Jacobi operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Let 0 < γ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' We consider the non-homogeneous time-fractional pseudo-parabolic equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1) Dγ 0+,t (u(t, x) − a∆α,βu(t, x)) − ∆α,βu(t, x) + mu(t, x) = f(t, x), (t, x) ∈ D, with initial condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2) u(0, x) = φ(x), x ∈ R+, where the functions f and φ are given functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Our aim is to find unique solution u of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1) - (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Let 0 < γ ≤ 1 and λ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Suppose that f ∈ C1([0, T], L2(µ)) and φ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Then the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2) has a unique solution u ∈ Cγ([0, T], L2(µ)) ∩ C([0, T], H) and can be represented by formula u(t, x) = � ∞ 0 � ∞ 0 � t 0 (t − τ)γ−1Eγ,γ � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)(t − τ)γ � f(τ, y) 1 + a(λ2 + ρ2) × ϕα,β λ (y)ϕα,β λ (x)dτdµα,β(y)dνα,β(λ) + � ∞ 0 � ∞ 0 Eγ,1 � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)tγ � φ(y)ϕα,β λ (y)ϕα,β λ (x)dµα,β(y)dνα,β(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' We assume that 0 < γ ≤ 1, λ ∈ R+ and u(t, ·) ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' We first prove that the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2) has only one solution, if the later exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Suppose the proposi- tion were false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Assume that there exist two different solutions u1(t, x) and u2(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Denote u0(t, x) = u1(t, x) − u2(t, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Then u0(t, x) solves the following equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='3) Dγ 0+,t (u0(t, x) − a∆α,βu0(t, x)) − ∆α,βu0(t, x) + mu0(t, x) = 0, (t, x) ∈ D, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='4) u0(0, x) = 0, x ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='3)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='4) has only trivial solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' This implies uniqueness of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Now, we will prove the existence of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Using the Fourier-Jacobi trans- form Fα,β (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2) on both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2), we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='5) Dγ 0+,t�u(t, λ) + λ2 + ρ2 + m 1 + a(λ2 + ρ2)�u(t, λ) = �f(t, λ) 1 + a(λ2 + ρ2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='6) �u(0, λ) = �φ(λ), for all λ ∈ R+ and 0 < t < T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The solution (see [KST06, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 231, ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='9]) of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='5)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='6) is given by 10 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' BEKBOLAT AND N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' TOKMAGAMBETOV (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='7) �u(t, λ) = � t 0 (t − τ)γ−1Eγ,γ � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)(t − τ)γ � �f(τ, λ) 1 + a(λ2 + ρ2)dτ + �φ(λ)Eγ,1 � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)tγ � , where Eγ,1(z) is the classical Mittag-Leffler function and Eγ,γ(z) is the Mittag-Leffler type function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Now by using the inverse Fourier-Jacobi transform F −1 α,β (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='3) to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='7), we obtain the formula for the solution of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2), given by u(t, x) = � ∞ 0 � ∞ 0 � t 0 (t − τ)γ−1Eγ,γ � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)(t − τ)γ � f(τ, y) 1 + a(λ2 + ρ2) × ϕα,β λ (y)ϕα,β λ (x)dτdµα,β(y)dνα,β(λ) + � ∞ 0 � ∞ 0 Eγ,1 � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)tγ � φ(y)ϕα,β λ (y)ϕα,β λ (x)dµα,β(y)dνα,β(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' By using the property d dτ (Eγ,1(cτ γ)) = cτ γ−1Eγ,γ(cτ γ), c = constant, of the Mittag-Leffler function, we obtain d dτ � Eγ,1 � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)(t − τ)γ �� = λ2 + ρ2 + m 1 + a(λ2 + ρ2)(t − τ)γ−1Eγ,γ � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)(t − τ)γ � and we can write (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='7) in a form �u(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' λ) = � t 0 (t − τ)γ−1Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='γ � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)(t − τ)γ � �f(τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' λ) 1 + a(λ2 + ρ2)dτ + �φ(λ)Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)tγ � = 1 λ2 + ρ2 + m � t 0 d dτ � Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)(t − τ)γ �� �f(τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' λ)dτ + �φ(λ)Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)tγ � = 1 λ2 + ρ2 + m �f(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' λ) − 1 λ2 + ρ2 + m �f(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' λ)Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)tγ � − 1 λ2 + ρ2 + m � t 0 Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)(t − τ)γ � d dτ �f(τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' λ)dτ + �φ(λ)Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)tγ � PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR 11 by using the rule integration by parts and Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Let 0 < γ < 1 and f ∈ C1([0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' T],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' L2(µ)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' φ ∈ H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' then we can estimate u as follows ∥u(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ·)∥2 H = � ∞ 0 ��(λ2 + ρ2)�u(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' λ) ��2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) ≲ � ∞ 0 �����(λ2 + ρ2) �f(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' λ) λ2 + ρ2 + m ����� 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) + � ∞ 0 �����(λ2 + ρ2) �f(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' λ) λ2 + ρ2 + mEγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)tγ ������ 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) + � ∞ 0 ���� λ2 + ρ2 λ2 + ρ2 + m � t 0 Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)(t − τ)γ � d dτ �f(τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' λ)dτ ���� 2 × dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) + � ∞ 0 ����(λ2 + ρ2)�φ(λ)Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)tγ ����� 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) ≲ � ∞ 0 ��� �f(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' λ) ��� 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) + � ∞ 0 ��� �f(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' λ) ��� 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) + � ∞ 0 �� t 0 ���� d dτ �f(τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' λ) ���� dτ �2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) + � ∞ 0 ���(λ2 + ρ2)�φ(λ) ��� 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) ≲ ∥f(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ·)∥2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='µ + ∥f(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ·)∥2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='µ + � T 0 ∥ d dtf(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ·)∥2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='µdt + ∥φ∥2 H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' here we have used Cauchy-Schwarz inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Fubibi’s theorem and a ≲ b denotes a ≤ cb for some positive constant c independent of a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Thus, ∥u(t, ·)∥2 H ≲ ∥f(t, ·)∥2 2,µ + ∥f(0, ·)∥2 2,µ + � T 0 ∥ d dtf(t, ·)∥2 2,µdt + ∥φ∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Then, we obtain ∥u∥2 C([0,T],H) ≲ ∥f∥2 C1([0,T],L2(µ)) + ∥φ∥2 H < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' In a case γ = 1, we have ∥u(t, ·)∥2 H = � ∞ 0 ��(λ2 + ρ2)�u(t, λ) ��2 dνα,β(λ) = � ∞ 0 �����(λ2 + ρ2) � t 0 �f(τ, λ) 1 + a(λ2 + ρ2)e − λ2+ρ2+m 1+a(λ2+ρ2) (t−τ)dτ + (λ2 + ρ2)�φ(λ)e − λ2+ρ2+m 1+a(λ2+ρ2) t ����� 2 dνα,β(λ) ≲ � ∞ 0 ���� � t 0 �f(τ, λ)e − λ2+ρ2+m 1+a(λ2+ρ2) (t−τ)dτ ���� 2 dνα,β(λ) 12 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' BEKBOLAT AND N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' TOKMAGAMBETOV + � ∞ 0 ����(λ2 + ρ2)�φ(λ)e − λ2+ρ2+m 1+a(λ2+ρ2) t ���� 2 dνα,β(λ) ≲ � ∞ 0 � T 0 ��� �f(t, λ) ��� 2 dtdνα,β(λ) + � ∞ 0 ���(λ2 + ρ2)�φ(λ) ��� 2 dνα,β(λ) = � T 0 ∥f(t, ·)∥2 2,µdt + ∥φ∥2 H by using Cauchy-Schwarz inequality and Fubini’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Thus, ∥u(t, ·)∥2 H ≲ � T 0 ∥f(t, ·)∥2 2,µdt + ∥φ∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Then, we have ∥u∥2 C([0,T],H) ≲ ∥f∥2 C([0,T],L2(µ)) + ∥φ∥2 H < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Let us estimate the function Dγ 0+,tu ∥Dγ 0+,tu(t, ·)∥2 2,µ = ∥Dγ 0+,t�u(t, ·)∥2 2,ν = � ∞ 0 ���Dγ 0+,t�u(t, ·) ��� 2 dνα,β(λ) = � ∞ 0 ����� �f(t, λ) 1 + a(λ2 + ρ2) − λ2 + ρ2 + m 1 + a(λ2 + ρ2)�u(t, λ) ����� 2 dνα,β(λ) ≲ ∥ �f(t, ·)∥2 2,ν + ∥�u(t, ·)∥2 2,ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Thus, we have ∥Dγ 0+,tu(t, ·)∥2 2,µ ≲ ∥f(t, ·)∥2 2,µ + ∥u(t, ·)∥2 2,µ and ∥Dγ 0+,tu∥2 C([0,T],L2(µ)) ≲ ∥f∥2 C([0,T],L2(µ)) + ∥u∥2 C([0,T],L2(µ)) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Consequently, using Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='10 we obtain u ∈ Cγ([0, T], L2(µ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Our prove is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The ISP for the time-fractional pseudo-parabolic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' This subsec- tion deals with the ISP for the time-fractional pseudo-parabolic equation associated with the Jacobi operator ∆α,β (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Statement of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Let 0 < γ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' We aim to find a couple of functions (u, f) satisfying equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='8) Dγ 0+,t (u(t, x) − a∆α,βu(t, x)) − ∆α,βu(t, x) + mu(t, x) = f(x), (t, x) ∈ D, under conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='9) u(0, x) = φ(x), x ∈ R+, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='10) u(T, x) = ψ(x), x ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR 13 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Let 0 < γ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Assume that ψ, φ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Then the pair (u, f) is a unique solution of the ISP (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='8)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' which are functions u ∈ Cγ([0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' T],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' L2(µ)) ∩ C([0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' T],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' H),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' f ∈ L2(µ) can be represented by the formulas u(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' x) = � ∞ 0 � ∞ 0 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)tγ� 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ �ψ(y)ϕα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β λ (y)ϕα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β λ (x)dµα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(y)dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) − � ∞ 0 � ∞ 0 Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ� − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)tγ� 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � × φ(y)ϕα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β λ (y)ϕα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β λ (x)dµα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(y)dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) and f(x) = � ∞ 0 � ∞ 0 (λ2 + ρ2 + m) ψ(y) − φ(y)Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ� 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � × ϕα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β λ (y)ϕα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β λ (x)dµα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(y)dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' We assume that 0 < γ ≤ 1, and u(t, ·), f ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Let us first prove the existence result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' By using the Fourier-Jacobi transform Fα,β (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2) on both sides of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='8)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='10), we obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='11) Dγ 0+,t�u(t, λ) + λ2 + ρ2 + m 1 + a(λ2 + ρ2)�u(t, λ) = �f(λ) 1 + a(λ2 + ρ2), (t, λ) ∈ D, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='12) �u(0, λ) = �φ(λ), λ ∈ R+, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='13) �u(T, λ) = �ψ(λ), λ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Solution of the equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='11) is given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='14) �u(t, λ) = �f(λ) λ2 + ρ2 + m + C(λ)Eγ,1 � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)tγ � , for all 0 < γ ≤ 1 and functions �f(λ) and C(λ) are unknown functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' For determine these functions we use conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='12) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' After that we have �u(0, λ) = �f(λ) λ2 + ρ2 + m + C(λ) = �φ(λ) and �u(T, λ) = �f(λ) λ2 + ρ2 + m + C(λ)Eγ,1 � − λ2 + ρ2 + m 1 + a(λ2 + ρ2)T γ � = �ψ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Thus we have C(λ) = �φ(λ) − �ψ(λ) 1 − Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � 14 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' BEKBOLAT AND N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' TOKMAGAMBETOV and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='15) �f(λ) = (λ2 + ρ2 + m) �ψ(λ) − �φ(λ)Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ� 1 − Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Substituting the resulting functions C(λ) and �f(λ) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='14), we get �u(t, λ) = 1 − Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)tγ� 1 − Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � �ψ(λ) − Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ� − Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)tγ� 1 − Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � �φ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Therefore solution of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='11) - (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='13) is the pair (�u, �f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' We obtain solution of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='8)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='10) by applying the inverse Fourier-Jacobi transform F −1 α,β (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='3) to the functions �u and �f, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='16) u(t, x) = � ∞ 0 � ∞ 0 1 − Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)tγ� 1 − Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ �ψ(y)ϕα,β λ (y)ϕα,β λ (x)dµα,β(y)dνα,β(λ) − � ∞ 0 � ∞ 0 Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ� − Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)tγ� 1 − Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � × φ(y)ϕα,β λ (y)ϕα,β λ (x)dµα,β(y)dνα,β(λ) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='17) f(x) = � ∞ 0 � ∞ 0 (λ2 + ρ2 + m) ψ(y) − φ(y)Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ� 1 − Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � × ϕα,β λ (y)ϕα,β λ (x)dµα,β(y)dνα,β(λ), for all 0 < γ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Let ψ, φ ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Then using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='12 we can estimate the function u as following ∥u(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ·)∥2 H = � ∞ 0 |(λ2 + ρ2)�u(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' λ)|2dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) ≲ � ∞ 0 ������ (λ2 + ρ2) �ψ(λ) 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)tγ� 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � ������ 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR 15 + � ∞ 0 ������ (λ2 + ρ2)�φ(λ) Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ� − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)tγ� 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � ������ 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) ≲ � ∞ 0 |(λ2 + ρ2) �ψ(λ)|2dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) + � ∞ 0 |(λ2 + ρ2)�φ(λ)|2dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Thus, ∥u(t, ·)∥2 H ≲ ∥ψ∥2 H + ∥φ∥2 H < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Then we have ∥u∥C([0,T],H) ≲ ∥ψ∥H + ∥φ∥H < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Let us estimate the function f ∥f∥2 2,µ = ∥ �f∥2 2,ν = � ∞ 0 | �f(λ)|2dνα,β(λ) = � ∞ 0 ������ (λ2 + ρ2 + m) �ψ(λ) − �φ(λ)Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ� 1 − Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � ������ 2 dνα,β(λ) ≲ � ∞ 0 ������ (λ2 + ρ2 + m) �ψ(λ) 1 − Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � ������ 2 dνα,β(λ) + � ∞ 0 ������ (λ2 + ρ2 + m) �φ(λ)Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ� 1 − Eγ,1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � ������ 2 dνα,β(λ) ≲ ∥ψ∥2 H + ∥φ∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' So, we obtain ∥f∥2 2,µ ≲ ∥ψ∥2 H + ∥φ∥2 H < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Next, we estimate the function Dγ 0+,tu ∥Dγ 0+,tu(t, ·)∥2 2,µ = ∥Dγ 0+,t�u(t, ·)∥2 2,ν = � ∞ 0 |Dγ 0+,t�u(t, λ)|2dνα,β(λ) = � ∞ 0 ����� �f(λ) 1 + a(λ2 + ρ2) − λ2 + ρ2 + m 1 + a(λ2 + ρ2)�u(t, λ) ����� 2 dνα,β(λ) ≲ ∥f∥2 2,ν + ∥u(t, ·)∥2 2,ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Finally, we have ∥Dγ 0+,tu∥2 C([0,T],L2(µ)) ≲ ∥f∥2 2,µ + ∥u∥2 C([0,T],L2(µ)) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' It is obvious that ∥u∥2 C([0,T],L2(µ)) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The existence is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Now, let us prove the uniqueness of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Taking into account the property of the Fourier-Jacobi transform Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='4, one observes that a pair of functions (u, f) is uniquely determined by the formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='16) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' The uniqueness is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' □ 16 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' BEKBOLAT AND N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' TOKMAGAMBETOV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Stability Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Finally, we study a stability property of the solution (u, f) of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='8)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='10) given by the formulas (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='16) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='17), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Let (u, f) and (ud, fd) be solutions of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='8)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='10) corre- sponding to the data (φ, ψ) and its small perturbation (φd, ψd), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Then the solution of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='8)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='10) depends continuously on these data, namely, we have ∥u − ud∥2 C([0,T],H) ≲ ∥ψ − ψd∥2 H + ∥φ − φd∥2 H and ∥f − fd∥2 2,µ ≲ ∥ψ − ψd∥2 H + ∥φ − φd∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' From the definition of the Fourier-Jacobi transform (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2) Fα,β(u(t, ·))(λ) = �u(t, λ) = � ∞ 0 u(t, x)ϕα,β λ (x)dµα,β(x), we conclude that Fα,β(u(t, ·) − ud(t, ·))(λ) = � ∞ 0 (u(t, x) − ud(t, x))ϕα,β λ (x)dµα,β(x) = � ∞ 0 u(t, x)ϕα,β λ (x)dµα,β(x) − � ∞ 0 ud(t, x)ϕα,β λ (x)dµα,β(x) = Fα,β(u(t, ·))(λ) − Fα,β(ud(t, ·))(λ) = �u(t, λ) − �ud(t, λ), here we have used property of the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' According to the above statement and using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' we have ∥u(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ·) − ud(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ·)∥2 H = � ∞ 0 (λ2 + ρ2)2|�u(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' λ) − �ud(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' λ)|2dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) = � ∞ 0 (λ2 + ρ2)2 ����� 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)tγ� 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � � �ψ(λ) − �ψd(λ) � − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ� − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)tγ� 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � � �φ(λ) − �φd(λ) � ���� 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) ≲ � ∞ 0 (λ2 + ρ2)2 ��� �ψ(λ) − �ψd(λ) ��� 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) + � ∞ 0 (λ2 + ρ2)2 ����φ(λ) − �φd(λ) ��� 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) = ∥ψ − ψd∥2 H + ∥φ − φd∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Thus, one gets ∥u(t, ·) − ud(t, ·)∥2 H ≲ ∥ψ − ψd∥2 H + ∥φ − φd∥2 H, and ∥u − ud∥2 C([0,T],H) ≲ ∥ψ − ψd∥2 H + ∥φ − φd∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR 17 By writing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='15) in the form �f(λ) = λ2 + ρ2 + m 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � �ψ(λ) − (λ2 + ρ2 + m)Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ� 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � �φ(λ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' and applying similar estimates again we can observe that ∥f − fd∥2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='µ = ∥ �f − �fd∥2 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='ν = � ∞ 0 ��� �f(λ) − �fd(λ) ��� 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) = � ∞ 0 ����� λ2 + ρ2 + m 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � � �ψ(λ) − �ψd(λ) � − (λ2 + ρ2 + m)Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ� 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � � �φ(λ) − �φd(λ) ������ 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) ≲ � ∞ 0 ������ λ2 + ρ2 + m 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � � �ψ(λ) − �ψd(λ) � ������ 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) + � ∞ 0 ������ (λ2 + ρ2 + m)Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ� 1 − Eγ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='1 � − λ2+ρ2+m 1+a(λ2+ρ2)T γ � � �φ(λ) − �φd(λ) � ������ 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) ≲ � ∞ 0 (λ2 + ρ2)2 ��� �ψ(λ) − �ψd(λ) ��� 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) + � ∞ 0 (λ2 + ρ2)2 ����φ(λ) − �φd(λ) ��� 2 dνα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='β(λ) = ∥ψ − ψd∥2 H + ∥φ − φd∥2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' It follows easily that ∥f − fd∥2 2,µ ≲ ∥ψ − ψd∥2 H + ∥φ − φd∥2 H, ending the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Stability Test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Here to check Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='3 we consider a ISP for the heat equa- tion with one dimensional Sturm-Liouville operator (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='18) ut(t, x) − uxx(t, x) = f(x), 0 < t < 1, x > 0, with conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='19) u(0, x) = u(1, x) = 0, where we put T = γ = 1, α = β = −1 2, a = m = 0 and φ(x) = ψ(x) = 0 for all x > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Also, consider a perturbed problem with some noise uǫ t(t, x) − uǫ xx(t, x) = f ǫ(x), 0 < t < 1, x > 0, with conditions uǫ(0, x) = 0, and uǫ(1, x) = ǫ · e−x2, x > 0, 18 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' BEKBOLAT AND N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' TOKMAGAMBETOV and with additional information φǫ(x) = 0 and ψǫ(x) = ǫ · e−x2, where ǫ is a positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Then by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2, we have uǫ(t, x) = ǫ √π � ∞ 0 1 − e−λ2t 1 − e−λ2 e− λ2 4 cos(λx)dλ, and f ǫ(x) = ǫ √π � ∞ 0 λ2e− λ2 4 (1 − e−λ2) cos(λx)dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Illustrations of our calculations above are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' ǫ 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='02 ∥ψ − ψǫ∥2 H 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='0006 ∥u − uǫ∥2 C([0,1],H) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='0003 ∥f − f ǫ∥2 2,µ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='0474 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='041897 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='0004 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Stability Test Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Table 1 confirms that the solution of the problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='8)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='10) is continuously depending on the given data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Small changes in the given data imply small changes in (u, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Appendix Calculations in Table 1 are made by using Maple 2021 program with the following codes: psi := exp(−x2), hat(psi) := int � 1 √ 2π psi · cos(x · λ), x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='.∞ � , norm(psi) := 40 · int � 4 √ 2π λ4 · |hat(psi)|2, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='.∞ � , hat(u) := 1 − exp(−λ2 · t) 1 − exp(−λ2) hat(psi), norm(u) := int � 4 √ 2π λ4 · |hat(u)|2, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='.∞ � , lim t→1(norm(u)), hat(f) := λ2 · hat(psi) 1 − exp(−λ2), and norm(f) := int � 4 √ 2π |hat(f)|2, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='.∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' PSEUDO-PARABOLIC EQUATIONS ASSOCIATED WITH THE JACOBI OPERATOR 19 References [Bus95] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' Bushuyev, Global 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08193 Barcelona, Spain and Institute of Mathematics and Mathematical Modeling Almaty, Kazakhstan E-mail address tokmagambetov@crm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='cat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content=' tokmagambetov@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} +page_content='kz' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdAyT4oBgHgl3EQfs_ko/content/2301.00585v1.pdf'} diff --git a/MdE0T4oBgHgl3EQfjAG9/content/tmp_files/2301.02454v1.pdf.txt b/MdE0T4oBgHgl3EQfjAG9/content/tmp_files/2301.02454v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..94669a7e706404bd59fd08b3b3336e9d73f47bf6 --- /dev/null +++ b/MdE0T4oBgHgl3EQfjAG9/content/tmp_files/2301.02454v1.pdf.txt @@ -0,0 +1,590 @@ +Optimizing the generation of polarization +squeezed light in nonlinear optical fibers driven +by femtosecond pulses +A. V. ANDRIANOV,1 N. A. KALININ,1,2 A. A. SOROKIN, 1 +E. A. ANASHKINA,1,3 L. L. SÁNCHEZ-SOTO,2,4,* J. F. CORNEY,5 +AND G. LEUCHS1,2,6 +1Institute of Applied Physics of the Russian Academy of Sciences, Nizhny Novgorod 603950, Russia +2Max Planck Institute for the Science of Light, 91058 Erlangen, Germany +3Advanced School of General and Applied Physics, Lobachevsky State University of Nizhny Novgorod, +Nizhny Novgorod 603022, Russia +4Departamento de Óptica, Facultad de Física, Universidad Complutense, 28040 Madrid, Spain +5School of Mathematics and Physics, University of Queensland, Brisbane, Queensland 4072, Australia +6Department of Physics, University of Erlangen-Nuremberg, 91058 Erlangen, Germany +*lsanchez@fis.ucm.es +Abstract: +Bright squeezed light can be generated in optical fibers utilizing the Kerr effect +for ultrashort laser pulses. However, pulse propagation in a fiber is subject to nonconservative +effects that deteriorate the squeezing. Here, we analyze two-mode polarization squeezing, +which is SU(2)-invariant, robust against technical perturbations, and can be generated in a +polarization-maintaining fiber. We perform a rigorous numerical optimization of the process +and the pulse parameters using our advanced model of quantum pulse evolution in the fiber that +includes various nonconservative effects and real fiber data. Numerical results are consistent +with experimental results. +© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement +1. +Introduction +Squeezed light is one of the most important resources in quantum optics with many existing and +foreseen applications, including improving the sensitivity and precision of optical metrology, +quantum communications, and quantum computing with continuous variables [1–4]. Squeezed +light, as a theoretical concept, has been studied for a long time (see, e.g. [5] for a review). +However, the first experimental observation of squeezing was made in 1986 [6]. Since then, the +experimental methods have improved significantly, and quantum squeezing has already become +a useful technology for applications. For example, modern gravitational wave detectors use +squeezed light to enhance the sensitivity and increase the observable range in space [7]. Squeezed +light plays an important role in modern theoretical studies; e.g., quantum cavity electrodynamics, +quantum phase transitions [8], and symmetry breaking in quantum systems [9]. Squeezed light +can be generated by using various optical nonlinearities (see, e.g. [1] for a review), including +second-order nonlinear processes, such as parametric down-conversion and oscillations [10,11], +parametric up-conversion [12,13], third-order nonlinearity in atomic vapors and fibers, and also +by direct intensity noise reduction by driving semiconductors lasers with extremely low-noise +current source [14]. +In this work, we concentrate on the optical Kerr effect, which can produce squeezing in +amorphous media and it is not limited by phase-matching conditions, thus providing larger +bandwidth. In the simplest scheme, a coherent state of light is launched into the nonlinear +Kerr medium. Amplitude-phase correlations are induced because the nonlinear phase shift is +proportional to the intensity. These correlations result in the formation of a squeezed Wigner +arXiv:2301.02454v1 [quant-ph] 6 Jan 2023 + +distribution with elliptic contours in phase space, in contrast to the rotationally symmetric +Gaussian distribution of the initial coherent state. The full quantum treatment shows that the Kerr +interaction leads to a non-Gaussian periodic dynamics with the appearance of “cat states" and +recurrence to the initial coherent state [15,16]. However, for reasonable values of nonlinearities, +light power and loss-limited distances, the Gaussian approximation can be used within a large +margin. +In the first fiber experiment continuous-wave (CW) light was used, and less than 1dB squeezing +was achieved in 114-m long fiber [17]. This experiment required enormous efforts to overcome +destructive effects of losses and noise induced by Brillouin scattering on the thermally excited +guided acoustic phonons in the fiber (GAWBS–guided acoustic waves Brillouin scattering) +accumulated over the long fiber. It was then proposed to use pulsed light because it is much +easier to achieve high-peak power, keeping the average power at a moderate level, thus requiring +much shorter fibers and greatly reducing the effect of losses and GAWBS. Although most of +the following fiber squeezing studies rely on short pulses, we note that in modern fibers based +on glasses with very high nonlinearity and good transparency, e.g. chalcogenide and tellurite +glasses, CW or long-pulse squeezing may worth revisiting [18–20]. A quantum theory of pulse +propagation in dispersive nonlinear media [21] suggested that quadrature squeezing can be +achieved for pulsed light, especially for solitons that preserve their shape and peak intensity over +long distances despite dispersion. Early experiments utilized both nonsoliton [22,23] as well as +soliton pulse propagation [24,25]. +One obstacle in using Kerr squeezing is that the squeezed ellipse is tilted in phase space +with respect to the mean vector of the field amplitude so that the output quantum state is not +amplitude-squeezed, which hinders direct detection of the reduced noise with power detectors. +Several methods to overcome this obstacle were proposed, such as using reflection from a +highly dispersive cavity [17] or employing two-mode squeezing in Sagnac-type and Mach- +Zehnder-type fiber interferometers to facilitate heterodyne detection [22,24–28]. Symmetric +Sagnac interferometers [24,25] producing nearly vacuum squeezed state, as well as asymmetric +interferometers producing bright coherent squeezed states were used [27,29]. Another approach +relies on the spectral filtering of the pulse after nonlinear propagation, which converts noise +correlations between different spectral bands into directly detectable amplitude squeezing [30]. +One of the most robust techniques relies on squeezing of the uncertainty of the polarization state. +By generating two squeezed beams in two polarization modes of a polarization-maintaining +fiber and appropriately transforming the output polarization state, the reduced uncertainty of +the polarization state can be directly measured by power detectors [31–34]. The best squeezing +achieved so far with fibers was observed in such a system [32]. +Ultrashort pulses propagating in fibers are susceptible to nonconservative effects of spontaneous +and stimulated Raman scattering. It was quickly recognized [35] and tested in experiments and +simulations [32,33] that the Raman effect is one of the most important factors limiting squeezing +in optical fibers for ultrashort pulses. Whereas electronic Kerr nonlinearity is not sensitive to the +pulse duration, the delayed Raman contribution is. It is known that the influence of Raman on +the classical properties of ultrashort fiber solitons scales with the pulse duration, being much +more pronounced for shorter pulses. This suggests that increasing the pulse duration may also +help reduce detrimental Raman contribution to quantum squeezing. However, the comprehensive +analysis of pulsed Kerr squeezing and optimization over the full set of pulse parameters has not +been done yet. In this work we perform rigorous numerical simulations to test the dependence of +the squeezing on the pulse energy and pulse duration as well as the fiber length. We identified +the regions of optimum parameters. We also proposed simple analytical considerations that help +to identify the role of the Raman effect and obtain the approximate scaling of the optimal pulse +duration. The numerical results were supported by experimental data. + +2. +Polarization squeezing description and numerical modeling +We focus on two-mode polarization squeezing because its experimental realization is quite robust +and less susceptible to various technical disturbances. The scheme we consider both in our +modeling and experiment utilizes propagation of two pulses with the orthogonal polarizations +aligned along axes of a birefringent nonlinear fiber. Both pulses experience Kerr squeezing. The +polarization squeezing relies on the fact that the quantum uncertainty of the polarization state +of two properly combined Kerr squeezed states can in some direction be made smaller than the +shot- noise limit. Polarization state and polarization fluctuations can be described in terms of the +Stokes operators +ˆ𝑆0 = ˆ𝑎† +𝐻 ˆ𝑎𝐻 + ˆ𝑎† +𝑉 ˆ𝑎𝑉 , +ˆ𝑆1 = ˆ𝑎† +𝐻 ˆ𝑎𝐻 − ˆ𝑎† +𝑉 ˆ𝑎𝑉 , +(1) +ˆ𝑆2 = ˆ𝑎† +𝐻 ˆ𝑎𝑉 + ˆ𝑎† +𝑉 ˆ𝑎𝐻, +ˆ𝑆3 = 𝑖( ˆ𝑎† +𝑉 ˆ𝑎𝐻 − ˆ𝑎† +𝐻 ˆ𝑎𝑉 ), +where ˆ𝑎† +𝐻/𝑉 and ˆ𝑎𝐻/𝑉 are creation and annihilation operators of two field modes, corresponding +to orthogonal horizontal/vertical polarization modes. +The uncertainty relations for the polarization operator and the corresponding squeezing can +be defined in an SU(2)-invariant manner [36]. The operators ˆ𝑆1,2,3 can be represented as +Cartesian components of a Stokes operator vector ˆ𝑺 = ( ˆ𝑆1, ˆ𝑆2, ˆ𝑆3), and ˆ𝑆0 represents the total +photon number. We can define squeezing without explicit use of Cartesian projections of the +Stokes operator vector, by introducing the component 𝑆∥ parallel to the mean value ⟨ˆ𝑺⟩ and +two components ˆ𝑆⊥1, ˆ𝑆⊥2 in the plane orthogonal to ⟨ˆ𝑺⟩ (the so-called “dark plane") [33]. The +nontrivial uncertainty relation for variances Δ2 ˆ𝑆⊥1, Δ2 ˆ𝑆⊥2 then reads as Δ2 ˆ𝑆⊥1Δ2 ˆ𝑆⊥2 ≥ |⟨ ˆ𝑆∥⟩|2 . +The squeezing is observed if there are components in the dark plane that obey [36] +Δ2 ˆ𝑆⊥1 < |⟨ ˆ𝑆∥⟩| < Δ2 ˆ𝑆⊥2. +(2) +The SU(2) invariance implies that the rotations of the polarization states, which can be done +with the use of birefringent plates and polarization splitting and combining optics, do not destroy +the polarization squeezing, provided losses are small. Then the squeezing can be measured after +appropriate rotation of the polarization state and measurement of the Stokes parameter ˆ𝑆1 by +using a polarization splitter and a balanced detector [33]. Moreover, this quantum polarization +description can be mapped one-to-one onto the quantum description of SU(2) interferometers [37], +for which it is known that the sensitivity can be enhanced by using squeezed light states [38]. This +means that polarization squeezed light can be used for precision interferometric measurements. It +was shown that bright squeezed light can be used for increasing the precision of polarimetry [39], +and enhancing the sensitivity of polarization interferometer [40]. +Efficient numerical modeling of quantum dynamics leading to squeezed-state formation in +the fiber requires certain assumptions and simplifications. We assume that the pulses propagate +independently of each other in two polarization modes of the fiber. We apply the truncated Wigner +method to model the quantum dynamics. This method is based on reconstructing the Wigner +distribution by gathering a large number of stochastic trajectories using the stochastic nonlinear +Schrödinger equation [41–44]. Our particular implementation of this equation takes into account +fiber dispersion (up to the third order) and the nonlinear response mediated by both the Raman +and instantaneous electronic interactions. We model this equation with the parameters of a +particular fiber which was used in our experiment (second-order dispersion 𝛽2 = −10.5ps2/km, +third- order dispersion 𝛽3 = 0.155 ps3/km, nonlinear coefficient 𝛾 = 3 W−1km−1, and the Raman +response function as in [44]). The pulse parameters were chosen in the ranges covering the values +accessible in our experiment. We adopt the polarization squeezing detection scheme and used the +corresponding routine to calculate the squeezing from the numerically simulated data [33,44]. + +In numerical modeling we calculate the squeezing for various input pulse parameters and +various fiber lengths. We prepared initial conditions in the form of hyperbolic secant-shaped +pulses 𝐴 = 𝐴0/cosh (𝑡/𝜏) with different durations in the range 𝑇 = 0.11 − 0.5 ps (𝑇 is FWHM +duration, 𝑇 = 1.763𝜏) and with a pulse energy in the range 𝐸 = 22.5 − 120 pJ. We calculate the +quantum dynamics for a propagation distance of up to 30 m for each initial condition. For each +set of initial conditions, we modeled 5000 realizations of stochastic trajectories to reconstruct +the squeezing ellipse. In the process of modeling, we calculated the squeezing at intermediate +distances along the fiber and recorded the obtained values. After the calculation of squeezing, we +could also introduce the losses of the detection scheme, which are inevitable in the experiment. +3. +Experimental study +We carried out an experimental study of polarization squeezing, which allowed us to compare +the measurements with our theory. The experimental setup for generating and measuring +squeezing generally followed the schematic presented in [33]. The two-mode squeezed light +needed to achieve polarization squeezing was generated in the polarization-maintaining fiber +(3M FSPM-7811). Femtosecond pulses at the central wavelength of 1.56 𝜇m from a mode-locked +laser with an adjustable pulse width and energy were launched into both polarization axes with +equal power. The laser signal was shot-noise limited above radio frequencies of a few MHz. +Because of the fiber birrefringence and the difference in the group velocities of the polarization +modes, the pulses quickly separated in time and propagated in the fiber almost independently. +To match the pulse arrival time at the output we used two consecutive fiber pieces of precisely +equal length spliced together with swapped fast and slow axes (rotated by 90 degrees) [40]. This +allowed us to make the setup simple and robust eliminating the free-space interferometer required +in the original scheme [33] to adjust the pulse arrival time. We tested two fiber lengths of 5.2 and +30 meters. To measure squeezing we first adjusted the polarization state and orientation of the +squeezed ellipse using waveplates (as described in [33]) and measured the noise in the Stokes +parameter ˆ𝑆1 using a polarization beam splitter, a balanced photodetector and a radiofrequency +spectrum analyzer. We used several laser settings providing different pulse durations and for +each setting the pulse energy was optimized for the best squeezing. +4. +Results and analysis +The analysis of numerical data allowed us to identify the most important processes and parameters +affecting squeezing. The entire set of simulations provide a 3D data set, with squeezing calculated +as a function of pulse duration, pulse energy, and fibre length. The slices of the 3D data set +showing the squeezing versus input pulse duration and energy at eight distances along the fiber +are presented in Fig. 1. The simulation was carried out on a 14 × 14 grid, but we have used data +interpolation for a better visual representation. +We can see that at the very beginning of the pulse propagation the squeezing mainly depends +on the peak power of the pulse. This behavior is consistent with simple considerations: At small +distances the pulse shaping effects are not pronounced, so the pulse mainly experiences self-phase +modulation and hence acquires some squeezing proportional to the peak power and the fiber +length. To emphasize this, we add lines of constant peak power to the plot. At larger distances the +soliton effects begin to play an important role, so the pulse dynamics becomes more complicated. +Pronounced regions of better squeezing are formed along curved lines. Better squeezing is +observed for the pulse parameters close to the fundamental soliton. To demonstrate this, we +plot dashed black lines, corresponding to the soliton parameters 𝑇 = 1.763𝜏 = 3.526|𝛽2|/𝛾𝐸. +For two distances (7.2 m and 30 m) we also plot curves corresponding to the pulse energies +maximizing squeezing at variable pulse duration (dotted lines in Fig. 1). Note that for small and +intermediate fiber lengths and large durations, solitons do not have enough distance to form, so + +40 +60 +80 +100 +120 +0.2 +0.3 +0.4 +0.5 +40 +60 +80 +100 +120 +0.2 +0.3 +0.4 +0.5 +6.0 m +7.2 m +12.0 m +30.0 m +40 +60 +80 +100 +120 +0.2 +0.3 +0.4 +0.5 +Pulse duration, ps +0.6 m +40 +60 +80 +100 +120 +0.2 +0.3 +0.4 +0.5 +3.0 m +40 +60 +80 +100 +120 +Pulse energy, pJ +0.2 +0.3 +0.4 +0.5 +Pulse duration, ps +40 +60 +80 +100 +120 +Pulse energy, pJ +0.2 +0.3 +0.4 +0.5 +18.0 m +40 +60 +80 +100 +120 +Pulse energy, pJ +0.2 +0.3 +0.4 +0.5 +24.0 m +40 +60 +80 +100 +120 +Pulse energy, pJ +0.2 +0.3 +0.4 +0.5 +-20 +-16 +-12 +-8 +-4 +0 +Sqeezing, dB +Fig. 1. Simulated squeezing for different pulse energies and durations at eight distances +along the fiber from 0.6 to 30 meters (color maps). Dashed cyan lines in the plot for +0.6 m correspond to constant peak power. Dashed black lines in the rest of the plots +represent fundamental soliton parameters. Dotted lines in plots for 7.2 m and 30 m +correspond to the pulse energy maximizing squeezing for a given pulse duration. Black +dots show data points obtained in the experimental optimization of the pulse energy. +Measured squeezing values are shown next to each dot. +the squeezing for such pulses largely depends on the input pulse peak power. This results in a +C-shaped optimal energy curve for the fiber length of 7.2 m. +We also compared our numerical findings with experimental results. The experimental points, +obtained for several pulse durations after optimizing the pulse energy, are shown in Fig. 1 for the +fiber lengths of 7.2 m and 30 m. It is evident that the experimental points align very well with the +curves of optimum pulse energy obtained in numerical modeling. They represent two distinctive +cases. For shorter distances, the optimum pulse energy increases as the pulse duration increases +above ∼0.3 ps. For longer distances, the optimum pulse energy decreases as the pulse width +increases so that the pulse parameters stay close to those of fundamental soliton. The absolute +values of squeezing in the experiment are significantly smaller than the modeled ones, but this is +because the simulation presented in Fig. 1 does not include losses. However, the trends in the +optimal pulse parameters are similar, since the effect of losses on squeezing does not depend +directly on the pulse energy or duration. +We extracted the maximum squeezing and the corresponding pulse parameters for different +fiber lengths, as shown in Fig. 2. It is seen that better squeezing can be achieved in longer +fibers, requiring progressively larger pulse durations and lower pulse energies. From the data we +calculated the soliton number parameter 𝑁2 = 𝜏𝛾𝐸/2|𝛽2|, which characterizes how close the +pulse is to the fundamental soliton (𝑁=1 corresponds to the fundamental soliton). Note that at +short propagation distances the best squeezing is observed at energies of about 10% larger than +the soliton energy. For large distances the best squeezing is achieved for pulses very close to the +fundamental solitons. +Optical losses in the fiber, at the fiber output and in the squeezing detector can strongly limit +the squeezing, especially for very large values demonstrated in lossless modeling. The effect of +losses at the output and lower than unity efficiency of the detectors can be simply added on top of +the quantum dynamics modeling [45]. The effect of distributed fiber losses needs to be directly +modeled using the propagation equation, but it can also be included approximately as lumped +losses at the output, and we did so to speed up our modeling. In Fig. 2 we show the squeezing + +0 +10 +20 +30 +Fiber length, m +0.12 +0.16 +0.2 +0.24 +0.28 +0.32 +Pulse duration, ps +40 +60 +80 +100 +120 +Pulse energy, pJ +Soliton number N=100 +0 +10 +20 +30 +Fiber length, m +-20 +-16 +-12 +-8 +-4 +0 +Squeezing, dB +(a) +(b) +N=1 +Fig. 2. Maximum squeezing as a function of the fiber length for different losses (a): no +losses (black curve), intrinsic fiber losses only (red curve), fiber losses and external +losses of 5% (green curve), fiber losses and external losses of 20% (blue curve). Optimal +pulse parameters (b): energy (black curve, left axis), duration (blue curve, right axis), +soliton number (red curve, left axis, multiplied by 100). +calculated when different losses are included: intrinsic fiber losses of 1dB/km and losses at the +fiber output and in the detection scheme. Including only the fiber losses, the squeezing is still +very strong, although it starts to roll off with increasing fiber length. With other additional loss +of 20% (estimated value for our experiment) the observed squeezing saturates at around −6 dB, +which is close to the experimentally measured result. With smaller additional losses of 5%, +which seems feasible in a carefully optimized experiment, we can expect observed squeezing at +the level of −10 to −12 dB. +5. +Analysis of pulse duration limitations due to Raman effect +Now we discuss the optimization of the pulse duration. From the simple picture of the squeezing +building up due to the Kerr effect, one may expect that the squeezing would improve with +increasing soliton energy (and corresponding shortening of its durations). At small propagation +distances, regions of best squeezing are indeed observed for shorter durations and highest energies, +but the optimum is gradually shifted towards longer durations and smaller energies at larger +distances. For shorter pulses the squeezing degrades abruptly. We illustrate this in Fig. 3, in +which we plot the maximum achievable squeezing optimized with respect to the pulse energy as +a function of the pulse duration and the fiber length. A well-defined region of the best squeezing +is observed. The optimum pulse duration shifts slowly towards larger values as the propagation +distance increases. +The observed behavior can be explained by evaluating the influence of nonconservative Raman +effects. The Raman effect for ultrashort solitons manifests itself in gradual self-frequency shift +of the pulse central frequency [46]. The rate of soliton self-frequency shift is given by an +approximate formula [47] +𝑑Ω +𝑑𝑧 = 8𝑇𝑅|𝛽2| +15𝜏4 +, +(3) +where 𝑇𝑅 characterizes the strength of the Raman response, 𝑇𝑅 ∼ 3 − 4 fs depending on the +particular shape of the response function. For the quantum evolution the influence of the +Raman effect is more complicated, but some useful conclusions can be drawn based on the +following considerations. The squeezing is related to certain correlations between the frequency +side-bands of quantum noise. These correlations build up due to the Kerr effect during soliton +propagation. The Raman effect redistributes the spectral components of the soliton and destroys +these correlations. To be able to make an estimate, we assume that the correlations are destroyed + +0 +10 +20 +30 +Distance, m +0.2 +0.3 +0.4 +0.5 +Pulse duration, ps +K=0.2 +K=0.05 +K=0.02 +K=0.01 +-20 +-16 +-12 +-8 +-4 +0 +Sqeezing, dB +Fig. 3. Simulated squeezing optimized with respect to the pulse energy as a function +of the pulse duration and the fiber length. Black dashed lines correspond to constant +values of 𝐾 in (4). +and the squeezing is reduced when the soliton frequency spectrum is shifted by an amount +comparable to the pulse spectral width. The FWHM spectral width ΔΩ is inversely proportional +to the pulse duration ΔΩ ≈ 2/𝑇. This leads to the condition +|𝛽2|𝑇𝑅𝑧 +𝑇3 +≡ 𝐾 ≪ 1, +(4) +where the dimensionless coefficient 𝐾 absorbs all the constants. This condition must be well +fulfilled so that the Raman effects can be neglected. Figur 3 shows lines of constant 𝐾. If we +start from long pulse durations the squeezing starts to improve as the soliton duration decreases +for any fixed fiber length corresponding to increasing 𝐾. However, as 𝐾 increases too much the +squeezing saturates and then degrades. The contours of the best squeezing region coincide very +well with the analytical predictions. The threshold at which the Raman effect becomes important +corresponds to 𝐾 ∼ 0.05 . . . 0.1. Although the presented numerical modeling was carried out for +particular fiber parameters, the analytical condition (4) is fairly universal and thus could be used +as a guide in planning and optimizing experiments. +6. +Discussion and conclusion +Our numerical modeling provides useful insights in how to optimize fiber polarization squeezing. +The numerical results match the experimental observations fairly well in terms of the optimal +combinations of pulse energy and pulse duration, for short as well as for long fiber length. +However, the numerical results giving the best match were obtained for about fiber lengths +differing by about 30%. This can be explained by the fact that propagation in orthogonal +polarization modes is not completely independent. Near the input and output ends of the fiber +the pulses overlap in time, so the cross-Kerr interaction leads to an increase in the effective +nonlinearity experienced by the pulses. The distance at which the pulses separate in time is about +0.5 m. Along this distance, the cross-Kerr contribution induced by the orthogonal pulse with +the same energy and peak power is added to the selfaction of the considered pulse [47]. In our +simplified modeling we assumed independent propagation and neglected cross-Kerr effect, so +longer fiber length was required to achieve similar effect. +The experimentally measured squeezing is severely affected by losses. The squeezing saturates +as it approaches the limit set by losses in the fiber and the detection scheme. However, the +intrinsic fiber losses are quite low for considered fiber lengths and the theoretically achievable +squeezing is quite strong even with these internal losses taken into account. So, the modeling + +presented here shows that a significant increase of the observed squeezing is realistic for a new +experimental setup with largely reduced external losses. +In conclusion, we performed the numerical simulation of polarization quantum squeezing +in a nonlinear fiber aimed at the optimization of squeezing with respect to the pulse duration +and energy as well as the fiber length and losses. Based on the analysis of the 3D data space +obtained in the modeling we identified the parameter areas for the best squeezing and described +general trends covering a wide range of pulse and fiber parameters. We proposed a simple +analytical approximation, which takes into account the Raman effect and provides the optimal +pulse duration for given fiber parameters. +Funding. +Ministry of Science and Higher Education of the Russian Federation, contract 075-15-2022-316. +Disclosures. +The authors declare no conflicts of interest. +Data availability. +Data underlying the results presented in this paper are not publicly available at this +time but may be obtained from the authors upon reasonable request. +References +1. +U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, “30 years of squeezed light generation,” Phys. Scripta 91, +053001 (2016). +2. +A. I. Lvovsky, “Squeezed light,” (2014). +3. +H.-S. Zhong, H. Wang, Y.-H. Deng, M.-C. Chen, L.-C. Peng, Y.-H. Luo, J. Qin, D. Wu, X. Ding, Y. Hu, P. Hu, X.-Y. +Yang, W.-J. Zhang, H. Li, Y. Li, X. Jiang, L. Gan, G. Yang, L. You, Z. Wang, L. Li, N.-L. Liu, C.-Y. 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Grosz, “Master equation approach to propagation in nonlinear fibers,” Opt. +Lett. 46, 665–668 (2021). +44. A. A. Sorokin, E. A. Anashkina, J. F. Corney, V. Bobrovs, G. Leuchs, and A. V. Andrianov, “Numerical Simulations +on Polarization Quantum Noise Squeezing for Ultrashort Solitons in Optical Fiber with Enlarged Mode Field Area,” +Photonics 8, 226 (2021). +45. H.-A. Bachor and T. C. Ralph, A guide to experiments in quantum optics (Wiley-VCH, 2004), 2nd ed. +46. J. P. Gordon, “Theory of the soliton self-frequency shift,” Opt. Lett. 11, 662–664 (1986). +47. G. P. Agrawal, Nonlinear fiber optics (Academic Press, 2013), 5th ed. + diff --git a/MdE0T4oBgHgl3EQfjAG9/content/tmp_files/load_file.txt b/MdE0T4oBgHgl3EQfjAG9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3fa51d9a06e46f833ac638fa5755616bb8d83503 --- /dev/null +++ b/MdE0T4oBgHgl3EQfjAG9/content/tmp_files/load_file.txt @@ -0,0 +1,771 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf,len=770 +page_content='Optimizing the generation of polarization squeezed light in nonlinear optical fibers driven by femtosecond pulses A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' ANDRIANOV,1 N.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' SÁNCHEZ-SOTO,2,4,* J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' CORNEY,5 AND G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' LEUCHS1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='6 1Institute of Applied Physics of the Russian Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Nizhny Novgorod 603950,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Russia 2Max Planck Institute for the Science of Light,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 91058 Erlangen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Germany 3Advanced School of General and Applied Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Lobachevsky State University of Nizhny Novgorod,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Nizhny Novgorod 603022,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Russia 4Departamento de Óptica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Facultad de Física,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Universidad Complutense,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 28040 Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Spain 5School of Mathematics and Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' University of Queensland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Brisbane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Queensland 4072,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Australia 6Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' University of Erlangen-Nuremberg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 91058 Erlangen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Germany lsanchez@fis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='ucm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='es Abstract: Bright squeezed light can be generated in optical fibers utilizing the Kerr effect for ultrashort laser pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' However, pulse propagation in a fiber is subject to nonconservative effects that deteriorate the squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Here, we analyze two-mode polarization squeezing, which is SU(2)-invariant, robust against technical perturbations, and can be generated in a polarization-maintaining fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We perform a rigorous numerical optimization of the process and the pulse parameters using our advanced model of quantum pulse evolution in the fiber that includes various nonconservative effects and real fiber data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Numerical results are consistent with experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' © 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Introduction Squeezed light is one of the most important resources in quantum optics with many existing and foreseen applications, including improving the sensitivity and precision of optical metrology, quantum communications, and quantum computing with continuous variables [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Squeezed light, as a theoretical concept, has been studied for a long time (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' [5] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' However, the first experimental observation of squeezing was made in 1986 [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Since then, the experimental methods have improved significantly, and quantum squeezing has already become a useful technology for applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' For example, modern gravitational wave detectors use squeezed light to enhance the sensitivity and increase the observable range in space [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Squeezed light plays an important role in modern theoretical studies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=', quantum cavity electrodynamics, quantum phase transitions [8], and symmetry breaking in quantum systems [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Squeezed light can be generated by using various optical nonlinearities (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' [1] for a review), including second-order nonlinear processes, such as parametric down-conversion and oscillations [10,11], parametric up-conversion [12,13], third-order nonlinearity in atomic vapors and fibers, and also by direct intensity noise reduction by driving semiconductors lasers with extremely low-noise current source [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' In this work, we concentrate on the optical Kerr effect, which can produce squeezing in amorphous media and it is not limited by phase-matching conditions, thus providing larger bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' In the simplest scheme, a coherent state of light is launched into the nonlinear Kerr medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Amplitude-phase correlations are induced because the nonlinear phase shift is proportional to the intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' These correlations result in the formation of a squeezed Wigner arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='02454v1 [quant-ph] 6 Jan 2023 distribution with elliptic contours in phase space, in contrast to the rotationally symmetric Gaussian distribution of the initial coherent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The full quantum treatment shows that the Kerr interaction leads to a non-Gaussian periodic dynamics with the appearance of “cat states" and recurrence to the initial coherent state [15,16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' However, for reasonable values of nonlinearities, light power and loss-limited distances, the Gaussian approximation can be used within a large margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' In the first fiber experiment continuous-wave (CW) light was used, and less than 1dB squeezing was achieved in 114-m long fiber [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' This experiment required enormous efforts to overcome destructive effects of losses and noise induced by Brillouin scattering on the thermally excited guided acoustic phonons in the fiber (GAWBS–guided acoustic waves Brillouin scattering) accumulated over the long fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' It was then proposed to use pulsed light because it is much easier to achieve high-peak power, keeping the average power at a moderate level, thus requiring much shorter fibers and greatly reducing the effect of losses and GAWBS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Although most of the following fiber squeezing studies rely on short pulses, we note that in modern fibers based on glasses with very high nonlinearity and good transparency, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' chalcogenide and tellurite glasses, CW or long-pulse squeezing may worth revisiting [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' A quantum theory of pulse propagation in dispersive nonlinear media [21] suggested that quadrature squeezing can be achieved for pulsed light, especially for solitons that preserve their shape and peak intensity over long distances despite dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Early experiments utilized both nonsoliton [22,23] as well as soliton pulse propagation [24,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' One obstacle in using Kerr squeezing is that the squeezed ellipse is tilted in phase space with respect to the mean vector of the field amplitude so that the output quantum state is not amplitude-squeezed, which hinders direct detection of the reduced noise with power detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Several methods to overcome this obstacle were proposed, such as using reflection from a highly dispersive cavity [17] or employing two-mode squeezing in Sagnac-type and Mach- Zehnder-type fiber interferometers to facilitate heterodyne detection [22,24–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Symmetric Sagnac interferometers [24,25] producing nearly vacuum squeezed state, as well as asymmetric interferometers producing bright coherent squeezed states were used [27,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Another approach relies on the spectral filtering of the pulse after nonlinear propagation, which converts noise correlations between different spectral bands into directly detectable amplitude squeezing [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' One of the most robust techniques relies on squeezing of the uncertainty of the polarization state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' By generating two squeezed beams in two polarization modes of a polarization-maintaining fiber and appropriately transforming the output polarization state, the reduced uncertainty of the polarization state can be directly measured by power detectors [31–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The best squeezing achieved so far with fibers was observed in such a system [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Ultrashort pulses propagating in fibers are susceptible to nonconservative effects of spontaneous and stimulated Raman scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' It was quickly recognized [35] and tested in experiments and simulations [32,33] that the Raman effect is one of the most important factors limiting squeezing in optical fibers for ultrashort pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Whereas electronic Kerr nonlinearity is not sensitive to the pulse duration, the delayed Raman contribution is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' It is known that the influence of Raman on the classical properties of ultrashort fiber solitons scales with the pulse duration, being much more pronounced for shorter pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' This suggests that increasing the pulse duration may also help reduce detrimental Raman contribution to quantum squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' However, the comprehensive analysis of pulsed Kerr squeezing and optimization over the full set of pulse parameters has not been done yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' In this work we perform rigorous numerical simulations to test the dependence of the squeezing on the pulse energy and pulse duration as well as the fiber length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We identified the regions of optimum parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We also proposed simple analytical considerations that help to identify the role of the Raman effect and obtain the approximate scaling of the optimal pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The numerical results were supported by experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Polarization squeezing description and numerical modeling We focus on two-mode polarization squeezing because its experimental realization is quite robust and less susceptible to various technical disturbances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The scheme we consider both in our modeling and experiment utilizes propagation of two pulses with the orthogonal polarizations aligned along axes of a birefringent nonlinear fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Both pulses experience Kerr squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The polarization squeezing relies on the fact that the quantum uncertainty of the polarization state of two properly combined Kerr squeezed states can in some direction be made smaller than the shot- noise limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Polarization state and polarization fluctuations can be described in terms of the Stokes operators ˆ𝑆0 = ˆ𝑎† 𝐻 ˆ𝑎𝐻 + ˆ𝑎† 𝑉 ˆ𝑎𝑉 , ˆ𝑆1 = ˆ𝑎† 𝐻 ˆ𝑎𝐻 − ˆ𝑎† 𝑉 ˆ𝑎𝑉 , (1) ˆ𝑆2 = ˆ𝑎† 𝐻 ˆ𝑎𝑉 + ˆ𝑎† 𝑉 ˆ𝑎𝐻, ˆ𝑆3 = 𝑖( ˆ𝑎† 𝑉 ˆ𝑎𝐻 − ˆ𝑎† 𝐻 ˆ𝑎𝑉 ), where ˆ𝑎† 𝐻/𝑉 and ˆ𝑎𝐻/𝑉 are creation and annihilation operators of two field modes, corresponding to orthogonal horizontal/vertical polarization modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The uncertainty relations for the polarization operator and the corresponding squeezing can be defined in an SU(2)-invariant manner [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The operators ˆ𝑆1,2,3 can be represented as Cartesian components of a Stokes operator vector ˆ𝑺 = ( ˆ𝑆1, ˆ𝑆2, ˆ𝑆3), and ˆ𝑆0 represents the total photon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We can define squeezing without explicit use of Cartesian projections of the Stokes operator vector, by introducing the component 𝑆∥ parallel to the mean value ⟨ˆ𝑺⟩ and two components ˆ𝑆⊥1, ˆ𝑆⊥2 in the plane orthogonal to ⟨ˆ𝑺⟩ (the so-called “dark plane") [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The nontrivial uncertainty relation for variances Δ2 ˆ𝑆⊥1, Δ2 ˆ𝑆⊥2 then reads as Δ2 ˆ𝑆⊥1Δ2 ˆ𝑆⊥2 ≥ |⟨ ˆ𝑆∥⟩|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The squeezing is observed if there are components in the dark plane that obey [36] Δ2 ˆ𝑆⊥1 < |⟨ ˆ𝑆∥⟩| < Δ2 ˆ𝑆⊥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' (2) The SU(2) invariance implies that the rotations of the polarization states, which can be done with the use of birefringent plates and polarization splitting and combining optics, do not destroy the polarization squeezing, provided losses are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Then the squeezing can be measured after appropriate rotation of the polarization state and measurement of the Stokes parameter ˆ𝑆1 by using a polarization splitter and a balanced detector [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Moreover, this quantum polarization description can be mapped one-to-one onto the quantum description of SU(2) interferometers [37], for which it is known that the sensitivity can be enhanced by using squeezed light states [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' This means that polarization squeezed light can be used for precision interferometric measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' It was shown that bright squeezed light can be used for increasing the precision of polarimetry [39], and enhancing the sensitivity of polarization interferometer [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Efficient numerical modeling of quantum dynamics leading to squeezed-state formation in the fiber requires certain assumptions and simplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We assume that the pulses propagate independently of each other in two polarization modes of the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We apply the truncated Wigner method to model the quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' This method is based on reconstructing the Wigner distribution by gathering a large number of stochastic trajectories using the stochastic nonlinear Schrödinger equation [41–44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Our particular implementation of this equation takes into account fiber dispersion (up to the third order) and the nonlinear response mediated by both the Raman and instantaneous electronic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We model this equation with the parameters of a particular fiber which was used in our experiment (second-order dispersion 𝛽2 = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='5ps2/km, third- order dispersion 𝛽3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='155 ps3/km, nonlinear coefficient 𝛾 = 3 W−1km−1, and the Raman response function as in [44]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The pulse parameters were chosen in the ranges covering the values accessible in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We adopt the polarization squeezing detection scheme and used the corresponding routine to calculate the squeezing from the numerically simulated data [33,44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' In numerical modeling we calculate the squeezing for various input pulse parameters and various fiber lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We prepared initial conditions in the form of hyperbolic secant-shaped pulses 𝐴 = 𝐴0/cosh (𝑡/𝜏) with different durations in the range 𝑇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='11 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='5 ps (𝑇 is FWHM duration, 𝑇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='763𝜏) and with a pulse energy in the range 𝐸 = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='5 − 120 pJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We calculate the quantum dynamics for a propagation distance of up to 30 m for each initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' For each set of initial conditions, we modeled 5000 realizations of stochastic trajectories to reconstruct the squeezing ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' In the process of modeling, we calculated the squeezing at intermediate distances along the fiber and recorded the obtained values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' After the calculation of squeezing, we could also introduce the losses of the detection scheme, which are inevitable in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Experimental study We carried out an experimental study of polarization squeezing, which allowed us to compare the measurements with our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The experimental setup for generating and measuring squeezing generally followed the schematic presented in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The two-mode squeezed light needed to achieve polarization squeezing was generated in the polarization-maintaining fiber (3M FSPM-7811).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Femtosecond pulses at the central wavelength of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='56 𝜇m from a mode-locked laser with an adjustable pulse width and energy were launched into both polarization axes with equal power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The laser signal was shot-noise limited above radio frequencies of a few MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Because of the fiber birrefringence and the difference in the group velocities of the polarization modes, the pulses quickly separated in time and propagated in the fiber almost independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' To match the pulse arrival time at the output we used two consecutive fiber pieces of precisely equal length spliced together with swapped fast and slow axes (rotated by 90 degrees) [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' This allowed us to make the setup simple and robust eliminating the free-space interferometer required in the original scheme [33] to adjust the pulse arrival time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We tested two fiber lengths of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 and 30 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' To measure squeezing we first adjusted the polarization state and orientation of the squeezed ellipse using waveplates (as described in [33]) and measured the noise in the Stokes parameter ˆ𝑆1 using a polarization beam splitter, a balanced photodetector and a radiofrequency spectrum analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We used several laser settings providing different pulse durations and for each setting the pulse energy was optimized for the best squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Results and analysis The analysis of numerical data allowed us to identify the most important processes and parameters affecting squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The entire set of simulations provide a 3D data set, with squeezing calculated as a function of pulse duration, pulse energy, and fibre length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The slices of the 3D data set showing the squeezing versus input pulse duration and energy at eight distances along the fiber are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The simulation was carried out on a 14 × 14 grid, but we have used data interpolation for a better visual representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We can see that at the very beginning of the pulse propagation the squeezing mainly depends on the peak power of the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' This behavior is consistent with simple considerations: At small distances the pulse shaping effects are not pronounced, so the pulse mainly experiences self-phase modulation and hence acquires some squeezing proportional to the peak power and the fiber length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' To emphasize this, we add lines of constant peak power to the plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' At larger distances the soliton effects begin to play an important role, so the pulse dynamics becomes more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Pronounced regions of better squeezing are formed along curved lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Better squeezing is observed for the pulse parameters close to the fundamental soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' To demonstrate this, we plot dashed black lines, corresponding to the soliton parameters 𝑇 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='763𝜏 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='526|𝛽2|/𝛾𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' For two distances (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 m and 30 m) we also plot curves corresponding to the pulse energies maximizing squeezing at variable pulse duration (dotted lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Note that for small and intermediate fiber lengths and large durations, solitons do not have enough distance to form, so 40 60 80 100 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='5 40 60 80 100 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='0 m 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 m 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='0 m 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='0 m 40 60 80 100 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='5 Pulse duration, ps 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='6 m 40 60 80 100 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='0 m 40 60 80 100 120 Pulse energy, pJ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='5 Pulse duration, ps 40 60 80 100 120 Pulse energy, pJ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='0 m 40 60 80 100 120 Pulse energy, pJ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='5 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='0 m 40 60 80 100 120 Pulse energy, pJ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='5 20 16 12 8 4 0 Sqeezing, dB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Simulated squeezing for different pulse energies and durations at eight distances along the fiber from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='6 to 30 meters (color maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Dashed cyan lines in the plot for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='6 m correspond to constant peak power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Dashed black lines in the rest of the plots represent fundamental soliton parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Dotted lines in plots for 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 m and 30 m correspond to the pulse energy maximizing squeezing for a given pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Black dots show data points obtained in the experimental optimization of the pulse energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Measured squeezing values are shown next to each dot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' the squeezing for such pulses largely depends on the input pulse peak power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' This results in a C-shaped optimal energy curve for the fiber length of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We also compared our numerical findings with experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The experimental points, obtained for several pulse durations after optimizing the pulse energy, are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 1 for the fiber lengths of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 m and 30 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' It is evident that the experimental points align very well with the curves of optimum pulse energy obtained in numerical modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' They represent two distinctive cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' For shorter distances, the optimum pulse energy increases as the pulse duration increases above ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='3 ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' For longer distances, the optimum pulse energy decreases as the pulse width increases so that the pulse parameters stay close to those of fundamental soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The absolute values of squeezing in the experiment are significantly smaller than the modeled ones, but this is because the simulation presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 1 does not include losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' However, the trends in the optimal pulse parameters are similar, since the effect of losses on squeezing does not depend directly on the pulse energy or duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We extracted the maximum squeezing and the corresponding pulse parameters for different fiber lengths, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' It is seen that better squeezing can be achieved in longer fibers, requiring progressively larger pulse durations and lower pulse energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' From the data we calculated the soliton number parameter 𝑁2 = 𝜏𝛾𝐸/2|𝛽2|, which characterizes how close the pulse is to the fundamental soliton (𝑁=1 corresponds to the fundamental soliton).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Note that at short propagation distances the best squeezing is observed at energies of about 10% larger than the soliton energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' For large distances the best squeezing is achieved for pulses very close to the fundamental solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Optical losses in the fiber, at the fiber output and in the squeezing detector can strongly limit the squeezing, especially for very large values demonstrated in lossless modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The effect of losses at the output and lower than unity efficiency of the detectors can be simply added on top of the quantum dynamics modeling [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The effect of distributed fiber losses needs to be directly modeled using the propagation equation, but it can also be included approximately as lumped losses at the output, and we did so to speed up our modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 2 we show the squeezing 0 10 20 30 Fiber length, m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='32 Pulse duration, ps 40 60 80 100 120 Pulse energy, pJ Soliton number N=100 0 10 20 30 Fiber length, m 20 16 12 8 4 0 Squeezing, dB (a) (b) N=1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Maximum squeezing as a function of the fiber length for different losses (a): no losses (black curve), intrinsic fiber losses only (red curve), fiber losses and external losses of 5% (green curve), fiber losses and external losses of 20% (blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Optimal pulse parameters (b): energy (black curve, left axis), duration (blue curve, right axis), soliton number (red curve, left axis, multiplied by 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' calculated when different losses are included: intrinsic fiber losses of 1dB/km and losses at the fiber output and in the detection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Including only the fiber losses, the squeezing is still very strong, although it starts to roll off with increasing fiber length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' With other additional loss of 20% (estimated value for our experiment) the observed squeezing saturates at around −6 dB, which is close to the experimentally measured result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' With smaller additional losses of 5%, which seems feasible in a carefully optimized experiment, we can expect observed squeezing at the level of −10 to −12 dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Analysis of pulse duration limitations due to Raman effect Now we discuss the optimization of the pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' From the simple picture of the squeezing building up due to the Kerr effect, one may expect that the squeezing would improve with increasing soliton energy (and corresponding shortening of its durations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' At small propagation distances, regions of best squeezing are indeed observed for shorter durations and highest energies, but the optimum is gradually shifted towards longer durations and smaller energies at larger distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' For shorter pulses the squeezing degrades abruptly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We illustrate this in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 3, in which we plot the maximum achievable squeezing optimized with respect to the pulse energy as a function of the pulse duration and the fiber length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' A well-defined region of the best squeezing is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The optimum pulse duration shifts slowly towards larger values as the propagation distance increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The observed behavior can be explained by evaluating the influence of nonconservative Raman effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The Raman effect for ultrashort solitons manifests itself in gradual self-frequency shift of the pulse central frequency [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The rate of soliton self-frequency shift is given by an approximate formula [47] 𝑑Ω 𝑑𝑧 = 8𝑇𝑅|𝛽2| 15𝜏4 , (3) where 𝑇𝑅 characterizes the strength of the Raman response, 𝑇𝑅 ∼ 3 − 4 fs depending on the particular shape of the response function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' For the quantum evolution the influence of the Raman effect is more complicated, but some useful conclusions can be drawn based on the following considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The squeezing is related to certain correlations between the frequency side-bands of quantum noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' These correlations build up due to the Kerr effect during soliton propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The Raman effect redistributes the spectral components of the soliton and destroys these correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' To be able to make an estimate, we assume that the correlations are destroyed 0 10 20 30 Distance, m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='5 Pulse duration, ps K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='2 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='05 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='02 K=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='01 20 16 12 8 4 0 Sqeezing, dB Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Simulated squeezing optimized with respect to the pulse energy as a function of the pulse duration and the fiber length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Black dashed lines correspond to constant values of 𝐾 in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' and the squeezing is reduced when the soliton frequency spectrum is shifted by an amount comparable to the pulse spectral width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The FWHM spectral width ΔΩ is inversely proportional to the pulse duration ΔΩ ≈ 2/𝑇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' This leads to the condition |𝛽2|𝑇𝑅𝑧 𝑇3 ≡ 𝐾 ≪ 1, (4) where the dimensionless coefficient 𝐾 absorbs all the constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' This condition must be well fulfilled so that the Raman effects can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Figur 3 shows lines of constant 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' If we start from long pulse durations the squeezing starts to improve as the soliton duration decreases for any fixed fiber length corresponding to increasing 𝐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' However, as 𝐾 increases too much the squeezing saturates and then degrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The contours of the best squeezing region coincide very well with the analytical predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The threshold at which the Raman effect becomes important corresponds to 𝐾 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Although the presented numerical modeling was carried out for particular fiber parameters, the analytical condition (4) is fairly universal and thus could be used as a guide in planning and optimizing experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Discussion and conclusion Our numerical modeling provides useful insights in how to optimize fiber polarization squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The numerical results match the experimental observations fairly well in terms of the optimal combinations of pulse energy and pulse duration, for short as well as for long fiber length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' However, the numerical results giving the best match were obtained for about fiber lengths differing by about 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' This can be explained by the fact that propagation in orthogonal polarization modes is not completely independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Near the input and output ends of the fiber the pulses overlap in time, so the cross-Kerr interaction leads to an increase in the effective nonlinearity experienced by the pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The distance at which the pulses separate in time is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content='5 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Along this distance, the cross-Kerr contribution induced by the orthogonal pulse with the same energy and peak power is added to the selfaction of the considered pulse [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' In our simplified modeling we assumed independent propagation and neglected cross-Kerr effect, so longer fiber length was required to achieve similar effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The experimentally measured squeezing is severely affected by losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The squeezing saturates as it approaches the limit set by losses in the fiber and the detection scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' However, the intrinsic fiber losses are quite low for considered fiber lengths and the theoretically achievable squeezing is quite strong even with these internal losses taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' So, the modeling presented here shows that a significant increase of the observed squeezing is realistic for a new experimental setup with largely reduced external losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' In conclusion, we performed the numerical simulation of polarization quantum squeezing in a nonlinear fiber aimed at the optimization of squeezing with respect to the pulse duration and energy as well as the fiber length and losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Based on the analysis of the 3D data space obtained in the modeling we identified the parameter areas for the best squeezing and described general trends covering a wide range of pulse and fiber parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' We proposed a simple analytical approximation, which takes into account the Raman effect and provides the optimal pulse duration for given fiber parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Funding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Ministry of Science and Higher Education of the Russian Federation, contract 075-15-2022-316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Disclosures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' The authors declare no conflicts of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Data availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE0T4oBgHgl3EQfjAG9/content/2301.02454v1.pdf'} +page_content=' L.' metadata={'source': 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100644 index 0000000000000000000000000000000000000000..3e70bd526c2f6e115b58f86c0b4fc6435670ed23 --- /dev/null +++ b/QdE1T4oBgHgl3EQfHQMq/content/tmp_files/2301.02923v1.pdf.txt @@ -0,0 +1,4965 @@ +arXiv:2301.02923v1 [math.AP] 7 Jan 2023 +NONLOCAL BOUNDARY-VALUE PROBLEMS WITH LOCAL BOUNDARY +CONDITIONS +JAMES M. SCOTT AND QIANG DU +Abstract. We describe and analyze nonlocal integro-differential equations with classical local +boundary conditions. The interaction kernel of the nonlocal operator has horizon parameter +dependent on position in the domain, and vanishes as the boundary of the domain is approached. +This heterogeneous localization allows for boundary values to be captured in the trace sense. +We state and prove a nonlocal Green’s identity for these nonlocal operators that involve local +boundary terms. We use this identity to state and establish the well-posedness of variational +formulations of the nonlocal problems with several types of classical boundary conditions. +We show the consistency of these nonlocal boundary-value problems with their classical local +counterparts in the vanishing horizon limit via the convergence of solutions. The Poisson data +for the local boundary-value problem is permitted to be quite irregular, belonging to the dual of +the classical Sobolev space. Heterogeneously mollifying this Poisson data for the local problem +on the same length scale as the horizon and using the regularity of the interaction kernel, we +show that the solutions to the nonlocal boundary-value problem with the mollified Poisson data +actually belong to the classical Sobolev space, and converge weakly to the unique variational +solution of the classical Poisson problem with original Poisson data. +1. Introduction +There has been much recent interest in nonlocal problems on a bounded domain Ω ⊂ Rd: +(1.1) +Lu = f in Ω , +associated to a nonlocal integral operator +(1.2) +Lu(x) := 2 +ˆ +Rd γ(x, y)(u(x) − u(y)) dy , +defined for measurable functions u : Rd → R and a nonlocal, symmetric and (often) nonnegative +kernel γ. These operators appear widely in both analysis and applications [2,4,5,8,9,13–15,17, +19,23,24,32,39,40,45,48,49,51,53,59]. Here, symmetry means γ(x, y) = γ(y, x), so that we can +identify L with a variational form associated with a quadratic nonlocal energy. Earlier studies +of variational formulations of (1.1) on bounded domains have taken several different paths. +Along the path that γ = γ(x, y) has a singularity at the diagonal x = y, both volume- +constraint problems and classical boundary-value problems have been investigated, see for ex- +ample [3,18,30,52] and additional references cited therein. If L defines a hypersingular integral +operator, in particular, then classical boundary values can be prescribed via the trace operator, +see [1, 50] for the cases of singular kernels that give rise to solutions in fractional Sobolev- +Slobodeckij spaces. +Down another path, γ = γ(x, y) is a compactly supported and translation invariant kernel, +e.g., γ(x, y) = δ−d−2ω(|x − y|/δ) for a function ω supported in the unit interval (0, 1) and a +constant (horizon parameter δ) that measures the range of nonlocal interactions. One natural +route to take is to define the so-called nonlocal volumetric constraint to complement the equation +2020 Mathematics Subject Classification. 45K05, 35J20, 46E35. +Key words and phrases. nonlocal equations, boundary-value problems, Poisson problem, Green’s identity, +nonlocal operators, heterogeneous localization, vanishing horizon. +This work is supported in part by the NSF DMS-1937254 and DMS-2012562. +1 + +2 +JAMES M. SCOTT AND QIANG DU +defined on Ω [24,25,28]. An example is the prescription of u(x) in a layer consisting of x ∈ Ωc +with dist(x, Ω) < δ. An alternative is to modify the nonlocal interaction rules involving u = u(x) +in a layered domain, say, for x ∈ Ω with dist(x, ∂Ω) < δ. These volumetric conditions can +recover traditional boundary conditions in the local limit as δ → 0 under suitable conditions, +see e.g., [6, 22, 28, 35, 44, 47]. Meanwhile, in the δ → ∞ limit with a rescaled fractional kernel, +these problems are related to studies of fractional differential equations defined on a bounded +domain [7,20,33,41]. In addition, one can find connections to the continuum limits of discrete +graph operators and discrete particle interactions [8,17,36]. +For various nonlocal problems, studies of their well-posedness subject to nonlocal volumetric +constraints can be found, for example, in [25,46], which offered desirable mathematical insight +as demonstrated for a number of applications such as the peridynamics models developed in +mechanics [16,29,43,54,55], nonlocal diffusion and jump processes [10,24] and nonlocal Stokes +equations for the analysis of smoothed particle hydrodynamics [27]. +Still another path is to mix classical boundary conditions and volume-constraint conditions +in constitutive models that blend local and nonlocal models. For an extensive discussion relating +to the many choices of blended models in applications such as peridynamics, see the survey [21]. +We are interested in boundary-value problems for nonlocal problems on a bounded domain +in the classical sense, that is, the boundary conditions are prescribed on ∂Ω only. The motiva- +tion is two-fold: first, while the nonlocal constraints are natural, they are not perfect choices. +Theoretically, nonlocal constraints may raise unintended concerns about the regularity of so- +lutions, for instance, non-constant functions vanishing in a layer of nonzero measure no longer +enjoy analyticity, and solutions of problems with smooth kernels may experience non-physical +or undesirable jumps at the boundary due to unmatched nonlocal constraints [28]. In practice, +developers of simulation codes for applications of nonlocal models, have ample practical reasons +to keep local boundary conditions in implementation even though a nonlocal model might be +derived and/or deemed a better modeling choice in the domain of interest. +To allow for the prescription of local boundary conditions, the nonlocal operator L and +the nonlocal solution spaces must be defined so that boundary values of the solutions make +sense. For a nonlocal operator associated with a kernel γ = γ(x, y) that does not have sufficient +singularity on the diagonal x = y, it means that some localizing property near the boundary +should hold. For instance, in [26,57,58], with a pair of constants β ∈ (0, 1) and δ > 0, a function +hδ(x) = min{δ, β dist(x, ∂Ω)} is introduced to characterize the extent of nonlocal interactions +at a point x ∈ Ω, instead of taking a constant δ as the horizon parameter everywhere in the +domain. A kernel similar to +γ(x, y) = +1 +2hδ(x)d+2 1{|x−y| 0, ∀r ∈ [0, r0]. +We also assume the normalization condition +(A2) +ˆ +B(0,1) +|z|2ρ(|z|) dz = d , +with B(0, 1) denoting the unit ball centered at the origin in Rd. +The function ηδ represents the extent of interaction that takes on a form in this paper +more complex than being merely a constant function. In the latter case, for a constant horizon +parameter δ > 0 we have ηδ(x) = δ for any x, With such a choice, the normalization condition +(A2) implies that for Ω = Rd, we have Lδv = ∆v for any quadratic function v = v(x). Indeed, +the choice of α = 2 is made in the definition of Lδ so that the nonlocal operator Lδ serves as +an analogue of classical second-order elliptic partial differential operator. Moreover, we consider +the case that ηδ = ηδ(x) is a smooth function in Ω and allows a specific form of heterogeneous +localization on the boundary ∂Ω, i.e., it is constant outside of a thin layer near ∂Ω and vanishes +as x approaches ∂Ω, as described below. +We make the following assumption on the domain Ω. +(AΩ) +Ω ⊂ Rd is a bounded C2 domain equipped with a heterogeneous localization +function ηδ i.e., there exist positive constants κ0 ∈ (0, 1), κ1, κ2 and ¯κ0 +depending only at most on d and Ω and a function ηδ : Ω → [0, ∞) +such that the heterogeneous localization properties (Aη) hold. + +4 +JAMES M. SCOTT AND QIANG DU +Here, the heterogeneous localization properties refer to +(Aη) +For every 0 < δ < δ0, where δ0 := min +� +1, 1 +9κ2 +1 +, 1 +2¯κ2 +0 +� +, ηδ satisfies +i) ηδ(x) = (dist(x, ∂Ω))2 for all x ∈ Ω with (dist(x, ∂Ω))2 < κ0δ, +ii) ηδ(x) = δ for all x ∈ Ω with κ0(dist(x, ∂Ω))2 > δ, +iii) ηδ ∈ C2(Ω) with |∇ηδ(x)| ≤ κ1 +√ +δ and |∇2ηδ(x)| ≤ κ2 for all x ∈ Ω, +iv) ηδ(x) ≤ ¯κ0 min{δ, (dist(x, ∂Ω))2} for all x ∈ Ω. +Note that for any C2 domain a heterogeneous localization function ηδ is guaranteed to +exist. +Indeed, one choice of ηδ is the following. +Choose κ0 ∈ (0, 1) to be small enough so +that the function x �→ dist(x, ∂Ω) belongs to C2({z ∈ Ω : dist(z, Ω) < √κ0}); such a κ0 will +always exist and depend at most on the boundary character of Ω and d, since Ω is a C2 domain. +For any δ > 0 construct via mollification a C∞ cutoff function ψδ satisfying ψδ ∈ C2(Ω), +ψδ ≡ 1 for κ0(dist(x, ∂Ω))2 > δ and ψδ = 0 for (dist(x, ∂Ω))2 < κ0δ with |∇ψδ(x)| ≤ C1 +√ +δ and +|∇2ψδ(x)| ≤ C2 +δ , where C1 and C2 depend only on d, Ω and κ0. Then we choose κ1 = C1 and +κ2 = C2, and define ηδ by +ηδ(x) = ψδ(x)δ + (1 − ψδ(x))(dist(x, ∂Ω))2 . +The δ0 defined above guarantees that that ηδ is in C2(Ω) for all δ < δ0. Therefore conditions +i)-iv) hold, where ¯κ0 = +2 +κ0 − 1. +This construction is of course not unique. One could choose a larger κ0 to ensure a wider +“transition region” {x ∈ Ω : κ0δ < (dist(x, ∂Ω))2 < κ−1 +0 δ}, which would require a small δ0 to be +chosen so that ηδ remains in C2(Ω) for all δ < δ0. +Associated to the nonlocal operator (1.3) and the nonlocal kernel ρδ,2 we define a real +symmetric bilinear form +Bρ,δ(u, v) := +ˆ +Ω +ˆ +Ω +ρδ,2(x, y) +� +u(x) − u(y) +�� +v(x) − v(y) +� +dy dx , +and so Hilbert space methods are available for the solvability of the nonlocal boundary-value +problem (1.1) subject to various local boundary conditions. +Thanks to the superlinear rate of localization at the boundary of Ω, we are able to establish +a nonlocal Green’s identity for our nonlocal operator, as formulated in the following theorem, +which serves as the foundation for consistent formulations of nonlocal boundary value problems. +Theorem 1.1. Let Ω ⊂ Rd satisfy (AΩ). Let ρ satisfy (A1)-(A2). Suppose u, v ∈ C2(Ω). Then +(1.5) +Bρ,δ(u, v) = +ˆ +Ω +Lδu(x) · v(x) dx + +ˆ +∂Ω +∂u +∂ν (x) · v(x) dσ(x) , +where ∂u +∂ν denotes the outward normal derivative of u. +We note that establishing (1.5) requires exact parametrizations of the domain near the +boundary so that the leading order corresponding integrals contained in the identity can be +computed. The proof is elaborate even for smooth functions u and v, which is shown first in +Section 3.3. Then, the identity is proved in Section 4.2 for more general functions based on +further characterizations of the nonlocal operator and nonlocal energy space. +Using the bilinear form Bρ,δ(u, v) we can introduce weak formulations of nonlocal problems +with classical boundary conditions by the following formal computation: Suppose that u is +smooth in Ω and satisfies the nonlocal Poisson problem for a given function f +(1.6) +Lδu = f , +in Ω . + +NONLOCAL BOUNDARY-VALUE PROBLEMS +5 +Then for an arbitrary smooth v, the nonlocal Green’s identity (1.5) gives +Bρ,δ(u, v) = +ˆ +Ω +f(x)v(x) dx + +ˆ +∂Ω +∂u +∂ν (x)v(x) dσ(x) . +The treatment of the boundary term depends on the type of boundary conditions considered +in the variational problem. In this work, for illustrative purposes, we treat Poisson problems +with two different classes of boundary conditions: prescribed Dirichlet data +(1.7) +u = g +on ∂Ω +and prescribed Neumann data +(1.8) +∂u +∂ν = g +on ∂Ω . +The energy space associated with the energy Bρ,δ(u, u) is a Hilbert space defined as +Wδ,2(Ω) := { closure of C∞(Ω) with respect to the norm ∥·∥Wδ,2(Ω)} , +and where the norm ∥·∥Wδ,2(Ω) is defined as +∥u∥2 +Wδ,2(Ω) := ∥u∥2 +L2(Ω) + [u]2 +Wδ,2(Ω) , +with the semi-norm defined, for a prescribed constant R ∈ (0, 1), by +(1.9) +[u]2 +Wδ,2(Ω) := +ˆ +Ω +ˆ +Ω∩{|y−x|≤Rηδ(x)} +��u(x) − u(y) +��2 +ηδ(x)d+2 +dy dx . +It turns out that the semi-norm [u]Wδ,2(Ω) is independent of the parameter R ∈ (0, 1) and +comparable to Bρ,δ(u, u) for any ρ satisfying (A1); see Theorem 1.6 and Theorem 1.7. The space +Wδ,2(Ω) inherits the properties of similar function spaces that have heterogeneous localization +at the boundary of Ω; for instance, the main results of [26, 58] hold for Wδ,2(Ω). Further, by +Theorem 1.5 +H1(Ω) ⊂ Wδ,2(Ω) ⊂ L2(Ω) . +Note that L2(Ω) and H1(Ω) (as well as H1 +0(Ω)) denote the conventional function spaces, see +Section 1.2 for other notation. +Equipped with sufficient knowledge of the nonlocal Hilbert space, we prove our next set of +main results: the variational formulation of each nonlocal boundary-value problem has a unique +finite-energy solution that belongs to a weakly closed subset of the nonlocal Hilbert space. These +well-posedness results for the four boundary-value problems are collected in Section 5, but we +state here the formal well-posedness result for the case of the nonlocal Poisson problem (1.6) +with the homogeneous Dirichlet boundary condition +(1.10) +u = 0 , +on ∂Ω . +Theorem 1.2. Let Ω satisfy (AΩ) and ρ satisfy (A1)-(A2). Then for each δ ∈ (0, δ0) there +exists a weakly closed subspace H of the nonlocal Hilbert space Wδ,2(Ω) with +H1 +0(Ω) ⊂ H ⊂ L2(Ω) , +such that the weak formulation of (1.6) and (1.10) is well-posed. That is, for every f ∈ H∗ there +exists a unique function u ∈ H such that +(1.11) +Bρ,δ(u, v) = ⟨f, v⟩ , +∀v ∈ H +satisfying the energy estimate +∥u∥H ≤ C(d, ρ, Ω) ∥f∥H∗ . + +6 +JAMES M. SCOTT AND QIANG DU +The weakly closed subset of the nonlocal Hilbert space varies depending on the type of +boundary conditions, that is, the choice of H varies from problem to problem. +Our third set of main results is the regularity of weak solutions to the well-posed nonlocal +problems. +The conclusion is formally summarized below, see Theorem 6.10 for the precise +statement. +Theorem 1.3. Let Ω ⊂ Rd satisfy (AΩ). Suppose that u ∈ Wδ,2(Ω) is a weak solution of (1.6)- +(1.10) i.e. u satisfies (1.11), where additionally f ∈ H1(Ω). Then there exists C depending only +on d, Ω and ρ such that +(1.12) +∥u∥H1(Ω) ≤ C +� +∥f∥H∗(Ω) + +���Φ−1 +δ,2f +��� +H1(Ω) +� +, +where Φδ,2(x) is defined as in (1.13). +The heart of the matter in the proof is to rewrite (1.6), valid for almost every x ∈ Ω, as a +pointwise relation +u(x) = Kδ,2u(x) + +f(x) +2Φδ,2(x) +for a.e. x ∈ Ω , +where the operator Kδ,α and the function Φδ,α for α ≥ 0 are defined as +(1.13) +Kδ,αu(x) := +1 +Φδ,α(x) +ˆ +Ω +ρδ,α(x, y)u(y) dy +and Φδ,α(x) := +ˆ +Ω +ρδ,α(x, y) dy , ∀x ∈ Ω . +We call the operators Kδ,α boundary-localized convolutions as they are of the convolution +type where ηδ is constant, which is the case away from the boundary, while the integral gets +localized at the boundary. Similar operators with mollification parameters that vanish at the +boundary were introduced in [11,12]. The particular form of operator we study in this work is +similar to the operators considered in [42], where boundedness in classical function spaces and +approximation properties were proved. With the estimates on the boundary-localized convolu- +tions in Section 6, one can use the smoothness of ρ and of f to show that u is in fact differentiable +thanks to the convolution structure of the right-hand side. +We note that the H1 regularity of the solutions to nonlocal problems with local boundary +conditions studied here extends to the whole domain Ω. This serves as a motivation behind +the use of local boundary conditions since this type of regularity does not hold in general +without heterogeneous localization. +The fact that smooth data leads to a smooth nonlocal +solution for the linear problems considered here highlights the difference from nonlocal boundary +conditions, where smooth data may still yield a nonsmooth nonlocal solution. The latter would +be nonphysical in general in the linear regime. +The regularity results allow us to establish the convergence of the unique variational solution +to (1.1) to the unique variational solution of (1.15) for both settings of boundary conditions, +which is our fourth set of main results. In fact, these convergence results remain true when the +Poisson data f in (1.15) belongs to the dual of the relevant Hilbert space. In the case of Dirichlet +data, f belongs to [H1 +0(Ω)]∗ = H−1(Ω), and in the case of Neumann boundary data f belongs +to a subspace of [H1(Ω)]∗. For both of the problems, we mollify the distribution f, so that the +regularized fδ belongs to a smaller dual space and thus allows for a well-posed nonlocal problem. +We then solve (1.1) with data fδ, and obtain convergence of solutions to the weak solution of +(1.15). The following is a formal statement in the case of homogeneous Dirichlet boundary data: +Theorem 1.4. Let Ω satisfy (AΩ) and ρ satisfy (A1)-(A2). Let f ∈ H−1(Ω) := [H1 +0(Ω)]∗ be a +given function. Then for each δ ∈ (0, δ0), if there exists a regularized approximation fδ ∈ H∗ of +f that satisfies the estimates (7.2) (see below) as well as +fδ ⇀ f weakly in H−1(Ω) , + +NONLOCAL BOUNDARY-VALUE PROBLEMS +7 +then the variational solutions uδ ∈ H to +(1.14) +Lδu = fδ , +in Ω , +and homogeneous Dirichlet boundary conditions (1.10) in fact belong to H1 +0(Ω) and converge +weakly in H1(Ω) as δ → 0 to a function u ∈ H1 +0(Ω). The function u is the unique variational +solution of the classical Poisson problem +(1.15) +− ∆u = f in Ω +with the homogeneneous Dirichlet boundary conditions (1.10). Moreover, for any f ∈ H−1(Ω) +such a sequence fδ is guaranteed to exist. +The uniform estimate (1.13), combined with the convergence results from Section 6 and +Section 7, ensure that the limit function u satisfies the classical Poisson equation. These conver- +gence results are stated precisely and proved for the general Dirichlet and Neumann problems +in Section 7. +To establish the main results as summarized above, we structure the rest of the paper as +follows: first, a brief introduction to the local and nonlocal function spaces is given before we end +Section 1. Section 2 contains technical details necessary for manipulating objects that depend +on ρ and ηδ. In Section 3, some properties of the nonlocal operator are studied in connection +to their actions on function spaces, and the nonlocal Green’s identity is shown for smooth +functions. The nonlocal function space is further analyzed in Section 4, providing necessary +tools for investigating boundary-value problems, including studies of the trace operator and the +proof of the nonlocal Green’s identity for wider classes of functions, together with the statements +of Poincaré inequalities. The variational formulations of nonlocal problems with local boundary +conditions and their well-posedness are then stated in Section 5. In Section 6, some integral +operators related to Lδ and their local limits are investigated, leading to the proofs of Poincaré +inequalities and regularity studies on the solution. We then examine the consistency with the +classical boundary-value problems in Section 7. Finally, in Section 8, we discuss various issues +such as the assumptions made in the current work, some possible extensions and future research +directions. +1.2. Notation and function spaces. For r > 0 and a bounded domain Ω ⊂ Rd, the open set +Ωr is defined as +Ωr := {x ∈ Ω : dist(x, ∂Ω) < r} . +The set of infinitely-differentiable functions with support compactly contained in Ω is de- +noted C∞ +c (Ω). The set of distributions is denoted D′(Ω). Lebesgue spaces are denoted Lp(Ω); +Sobolev spaces W k,p(Ω) for k ∈ N are defined via the integrability properties of weak derivatives +as in [31], with the convention W k,2(Ω) = Hk(Ω). The functional dual of a Banach space V is +denoted V ∗. +For functions u : Ω → R and v : Ω → R, we use ⟨u, v⟩ to denote the standard L2 inner +product on their domain of definition and also the duality pair where appropriate. The integral +average of u on Ω, or the duality pair with the constant function v ≡ 1/|Ω|, is denoted by +(u)Ω := + +Ω +u(x) dx := 1 +|Ω| +ˆ +Ω +u(x) dx = ⟨u, 1/|Ω|⟩ , +where |Ω| denotes the volume of Ω. +The space H1 +0(Ω) is the closure of C∞ +c (Ω) with respect to the H1(Ω) norm. We define +the subspace ˚ +H1(Ω) = {u ∈ H1(Ω) : (u)Ω = 0}. We use the common convention H−1(Ω) = +[H1 +0(Ω)]∗. + +8 +JAMES M. SCOTT AND QIANG DU +It is expected that the local space H1(Ω) is continuously embedded in the nonlocal energy +space Wδ,2(Ω). We state the result in the following theorem, which is analogous to similar results +given in [26,58] concerning nonlocal energy spaces involving heterogeneous localization. +Theorem 1.5 (Embedding). Let Ω ⊂ Rd satisfy (AΩ). Let ρ be a kernel satisfying (A1)-(A2). +Then there exists a constant C = C(Ω, δ0) such that for all δ < δ0 +[u]Wδ,2(Ω) ≤ C ∥∇u∥L2(Ω) +for all u ∈ H1(Ω). +Proof. It suffices to prove the estimate for u ∈ C∞(Ω). By the mean value theorem, +[u]2 +Wδ,2(Ω) = +ˆ +Ω +ˆ +|h|≤R0ηδ(x) +|u(x + h) − u(x)|2 +ηδ(x)d+2 +dh dx +≤ +ˆ +Ω +ˆ +|h|≤R0ηδ(x) +ˆ 1 +0 +|∇u(x + th)|2 dt +|h|2 +ηδ(x)d+2 dh dx += +ˆ +Ω +ˆ +|h|≤R0 +|h|2 +ˆ 1 +0 +|∇u(x + tηδ(x)h)|2 dt dh dx . +On the domain of integration the function gh(x) := x + tηδ(x)h satisfies +det ∇gh(x) = 1 + t∇ηδ(x) · h > 1 − tκ1R0 +√ +δ > 0 +for all δ < δ0. Therefore, the matrix inverse of ∇gh(x) is bounded from above. It follows that +[u]2 +Wδ,2(Ω) ≤ C(δ0) +ˆ +|h|≤R0 +|h|2 +ˆ +Ω +|∇u(x)|2 dx dh = C ∥∇u∥L2(Ω) . +□ +In a spirit similar to the studies in [26,58], we also have the independence of the nonlocal +energy norms on the scaling factor used in the horizon. +Theorem 1.6 (The nonlocal energy space is independent of the horizon). Let Ω ⊂ Rd satisfy +(AΩ). For R ∈ (0, 1), define the norm +∥u∥2 +Wδ,2(Ω),R := ∥u∥2 +L2(Ω) + [u]2 +Wδ,2(Ω),R , +where the semi-norm is defined as +[u]2 +Wδ,2(Ω),R := +ˆ +Ω +ˆ +Ω∩{|y−x|≤Rηδ(x)} +��u(x) − u(y) +��2 +ηδ(x)d+2 +dy dx . +Then, for constants 0 < r0 ≤ R0 < 1, there exists a constant C depending only on d, R0, r0 and +Ω such that +C−1 ∥u∥Wδ,2(Ω),R0 ≤ ∥u∥Wδ,2(Ω),r0 ≤ C ∥u∥Wδ,2(Ω),R0 +for any u ∈ C∞(Ω). +Proof. The proof follows essentially the same method from [58, Lemma 6.2] and [26, Lemma +2.2]. +□ +We now present the independence of the nonlocal energy norm space on the specific form +of the nonlocal kernel ρ so long the assumption (A1) holds. A by-product is the continuity of +the bilinear form. + +NONLOCAL BOUNDARY-VALUE PROBLEMS +9 +Theorem 1.7 (The nonlocal energy space is independent of the kernel). Let Ω ⊂ Rd satisfy +(AΩ) and ρ satisfy (A1). Then there exists a constant C depending only on d, ρ and Ω such that +(1.16) +C−1Bρ,δ(u, u) ≤ [u]Wδ,2(Ω) ≤ CBρ,δ(u, u) , +∀u ∈ Wδ,2(Ω) . +Moreover, we have the continuity of the bilinear form in the nonlocal energy space +(1.17) +Bρ,δ(u, v) ≤ C ∥u∥Wδ,2(Ω) ∥v∥Wδ,2(Ω) , +∀u, v ∈ Wδ,2(Ω). +Continuity also holds for Bρ,δ, where ρ is defined using ρ as in (2.8): +(1.18) +Bρ,δ(u, v) ≤ C ∥u∥Wδ,2(Ω) ∥v∥Wδ,2(Ω) , +∀u, v ∈ Wδ,2(Ω). +Proof. The first result of the theorem follows in a straightforward way from the upper and lower +bounds on ρ and Theorem 1.6. The inequality (1.17) then follows from Hölder’s inequality and +(1.16). The inequality (1.18) follows similarly. +□ +2. Properties of the heterogeneous localization function and the associated +kernels +We now present some properties related to the function ηδ and various kernels used in this +work. First, we note that any kernel satisfying (A2) also satisfies +(2.1) +ˆ +B(0,1) +(z ⊗ z)ρ(|z|) dz = I , +since for each pair of indices i, j, a coordinate change by the appropriate rotation gives +ˆ +B(0,1) +zizjρ(|z|) dz = δij +ˆ +B(0,1) +z2 +i ρ(|z|) dz = 1 +d +�ˆ +B(0,1) +|z|2ρ(|z|) dz +� +δij = δij . +In addition, we adopt the notation that for vectors a and b in Rd, a ⊗ b denotes the d × d +matrix with ij coordinate equal to aibj, and I denotes the d × d identity matrix. +2.1. Spatial variations of the heterogeneous localization function. For ease of access, we +record the following comparisons of the heterogeneous localization function ηδ that are frequently +referred to in later discussions: +Lemma 2.1. Let R0 ∈ (0, 1), and let Ω ⊂ Rd satisfy (AΩ). Then for all δ < δ0, +(2.2) +(1 − κ1R0 +√ +δ)ηδ(x) ≤ ηδ(y) ≤ (1 + κ1R0 +√ +δ)ηδ(x), +∀x, y ∈ Ω with |x − y| ≤ R0ηδ(x) +and +(2.3) (1 − κ1R0 +√ +δ)ηδ(y) ≤ ηδ(x) ≤ (1 + κ1R0 +√ +δ)ηδ(y), +∀x, y ∈ Ω with |x − y| ≤ R0ηδ(y) . +Proof. It suffices to show (2.2), since (2.3) will follow from the same arguments with the roles +of x and y interchanged. From the properties of ∇ηδ and Taylor expansion we get +ηδ(y) ≤ ηδ(x) + κ1 +√ +δ|x − y| ≤ (1 + κ1R0 +√ +δ)ηδ(x) +and +ηδ(x) ≤ ηδ(y) + κ1 +√ +δ|x − y| ≤ ηδ(y) + κ1R0 +√ +δηδ(x) . +□ + +10 +JAMES M. SCOTT AND QIANG DU +Lemma 2.2. Let R0 ∈ (0, 1). Let Ω ⊂ Rd satisfy (AΩ), and let r > 0 be such that Rd \ Ωr ̸= ∅. +Then for all δ < δ0 +(2.4) +{y : |x − y| ≤ R0ηδ(x)} ⊂ Ωr whenever dist(x, ∂Ω) < +r +1 + ¯κ0R0 +√ +δ +and +(2.5) +{y : |x − y| ≤ R0ηδ(y)} ⊂ Ωr whenever dist(x, ∂Ω) < (1 − ¯κ0R0 +√ +δ)r . +Also, +(2.6) +{y : |x − y| ≤ R0ηδ(x)} ⊂ Rd \ Ωr whenever dist(x, ∂Ω) ≥ +r +1 − ¯κ0R0 +√ +δ +and +(2.7) +{y : |x − y| ≤ R0ηδ(y)} ⊂ Rd \ Ωr whenever dist(x, ∂Ω) ≥ (1 + ¯κ0R0 +√ +δ)r . +Proof. For (2.4), since ηδ(x) ≤ ¯κ0 min{δ, (dist(x, ∂Ω))2} +dist(y, ∂Ω) ≤ dist(x, ∂Ω) + |x − y| ≤ dist(x, ∂Ω) + R0ηδ(x) +≤ dist(x, ∂Ω) + ¯κ0R0 +√ +δ dist(x, ∂Ω) , +and so +dist(y, ∂Ω) ≤ +� +1 + ¯κ0R0 +√ +δ +� +(dist(x, ∂Ω)) < r . +For (2.5), since ηδ(y) ≤ ¯κ0 min{δ, (dist(y, ∂Ω))2} +dist(y, ∂Ω) ≤ dist(x, ∂Ω) + |x − y| ≤ dist(x, ∂Ω) + R0ηδ(y) +≤ dist(x, ∂Ω) + ¯κ0R0 +√ +δ dist(y, ∂Ω) , +and so +(1 − ¯κ0R0 +√ +δ) dist(y, ∂Ω) ≤ dist(x, ∂Ω) ≤ (1 − ¯κ0R0 +√ +δ)r . +For (2.6), +dist(y, ∂Ω) ≥ dist(x, ∂Ω) − |y − x| ≥ dist(x, ∂Ω) − ¯κ0R0 min{δ, (dist(x, ∂Ω))2 +≥ dist(x, ∂Ω)(1 − ¯κ0R0 +√ +δ) , +and so +dist(y, ∂Ω) +(1 − ¯κ0R0 +√ +δ) +≥ dist(x, ∂Ω) ≥ +r +(1 − ¯κ0R0 +√ +δ) +. +For (2.7), +dist(y, ∂Ω) ≥ dist(x, ∂Ω) − |y − x| ≥ dist(x, ∂Ω) − ¯κ0R0 min{δ, (dist(y, ∂Ω))2} , +and so +(1 + ¯κ0R0 +√ +δ) dist(y, ∂Ω) ≥ dist(x, ∂Ω) ≥ (1 + ¯κ0R0 +√ +δ)r . +□ + +NONLOCAL BOUNDARY-VALUE PROBLEMS +11 +2.2. Auxiliary kernels. To study properties of nonlocal problems corresponding to operators +associated with the kernel ρ, such as the regularity of solutions, it is convenient to introduce an +additional kernel derived from ρ. We define ρ as +ρ(r) := −ρ′(r) +r +, +i.e. +ρ(r) = +ˆ 1 +r +sρ(s) ds . +(2.8) +Then by the assumptions on ρ, ρ is in C0([0, ∞)) with support in [0, R0], and satisfies +(2.9) +ˆ +B(0,1) +ρ(|z|) dz = ρd , where ρd := + + + + + +−2 +´ 1 +0 +ρ′(r) +r +dr , +d = 1 , +2πρ(0) , +d = 2 , +1 +d−2 +´ +B(0,1) +ρ(|z|) +|z|2 dz , +d ≥ 3 . +We also define the rescaled kernels +ρa,α(|z|) := +1 +ad+α ρ +�|z| +a +� +, for any quantity a > 0 and α ∈ R . +When α = 0 we write ρa(|z|) = ρa,0(|z|). We note the difference in the notation: while ρδ,α(|x − +y|) = δ−d−αρ(|x−y| +δ +), ρδ,α(x, y) is defined in (1.4). In fact, we have +2ρδ,α(x, y) = ρηδ(y),α(|y − x|) + ρηδ(x),α(|y − x|) = ρηδ(y)(|y − x|) +ηδ(y)α ++ ρηδ(x)(|y − x|) +ηδ(x)α +. +Likewise, we use ˜ρδ,α(x, y) and ˜ρδ,α(|x − y|) to denote the rescaled kernels corresponding +to a generic kernel ˜ρ. Consequently, ρδ,α(x, y) and ρδ,α(|x − y|) denote the rescaled kernels +corresponding to ρ. +The integrals of the kernels Φδ,α(x) are defined as in (1.13), and when α = 0 we write +Φδ(x) := Φδ,0(x). +2.3. The integral averages of the kernels. +Lemma 2.3. Suppose Ω ⊂ Rd satisfies (AΩ). Fix R0 ∈ (0, 1) and let �ρ be any function in +C0([0, ∞)) with support in [0, R0]. Then there exists a constant C depending only on d, �ρ, R0, +Ω and κ1 such that +(2.10) +ˆ +Ω +�ρηδ(x),α(|y − x|) dy ≤ +C +ηδ(x)α and +ˆ +Ω +�ρηδ(y),α(|y − x|) dy ≤ +C +ηδ(x)α +for any x ∈ Ω and any α ∈ R. +If in addition �ρ(r) ≥ ρ0 > 0 for all r ∈ [0, r0] for some r0 < R0, then +(2.11) +ˆ +Ω +�ρηδ(x),α(|y − x|) dy ≥ +1 +Cηδ(x)α and +ˆ +Ω +�ρηδ(y),α(|y − x|) dy ≥ +1 +Cηδ(x)α +for any x ∈ Ω and any α ∈ R. +Proof. Since supp �ρ ⊂ [0, R0] we obtain one of the lower bounds +ˆ +Ω +�ρηδ(x),α(|y − x|) dy ≤ ∥�ρ∥L∞([0,1]) +ˆ +{|y−x|≤R0ηδ(x)} +1 +ηδ(x)d+α dy = C(d, �ρ) +ηδ(x)α . +As for the upper bound on the same integral, since Ω satisfies +|B(x, r) ∩ Ω| ≥ Crd +for some constant C = C(d, Ω) for any x ∈ Ω and r < diam(Ω), +ˆ +Ω +�ρηδ(x),α(|y − x|) dy ≥ ρ0 +ˆ +Ω∩{|y−x|≤r0ηδ(x)} +1 +ηδ(x)d+α dy ≥ C(d, �ρ, Ω) +ηδ(x)α +. + +12 +JAMES M. SCOTT AND QIANG DU +Therefore we need only to prove the same inequalities for the other piece: +(2.12) +ˆ +Ω +�ρηδ(y),α(|y − x|) dy ≤ C(d, Ω, �ρ) +ηδ(x)α +and +(2.13) +ˆ +Ω +�ρηδ(y),α(|y − x|) dy ≥ C(d, Ω, �ρ) +ηδ(x)α +if �ρ ≥ ρ0 on [0, r0] . +Both of these inequalities follow in the same way using (2.3): +ˆ +Ω +�ρηδ(y),α(|y − x|) dy ≤ C +ˆ +{|y−x|≤C2R0ηδ(x)} +1 +ηδ(x)d+α �ρ +� |y − x| +C2ηδ(x) +� +dy = C(d, �ρ) +ηδ(x)α +and +ˆ +Ω +�ρηδ(y),α(|y − x|) dy ≥ Cρ0 +ˆ +Ω∩{|y−x|≤ r0 +C2 ηδ(x)} +1 +ηδ(x)d+α dy ≥ C(d, �ρ, Ω) +ηδ(x)α +. +□ +Corollary 2.4. With the assumptions of Lemma 2.3, for any α ≥ 0 and any β ∈ [0, α], there +exists a constant C(d, �ρ, α, β) > 0 such that +(2.14) +ˆ +Ω +|ηδ(y)|β �ρηδ(x),α(|y − x|) dy ≤ +C +ηδ(x)α−β and +ˆ +Ω +|ηδ(y)|β �ρηδ(y),α(|y − x|) dy ≤ +C +ηδ(x)α−β +and +(2.15) +ˆ +Ω +|ηδ(y)|β �ρηδ(x),α(|y − x|) dy ≥ +1 +Cηδ(x)α−β and +ˆ +Ω +|ηδ(y)|β �ρηδ(y),α(|y − x|) dy ≥ +1 +Cηδ(x)α−β +for any x ∈ Ω. +We note that for any ρ satisfying (A1), the estimates stated for �ρ in Lemma 2.3 and +Corollary 2.4 also hold for ρ. +From the definition (1.13), the normalization in (2.9), and Lemma 2.3, we also immediately +get the following. +Corollary 2.5. With the assumptions of Lemma 2.3, there exist constants µα and ¯µα depending +only on d, ρ, α and Ω such that +(2.16) +0 < µα ≤ |ηδ(x)|αΦδ,α(x) ≤ ¯µα < ∞, +∀x ∈ Ω . +2.4. Properties of kernel derivatives. +Theorem 2.6. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1), with ρ being defined as in (2.8). +Then for any α ∈ R, there exists C = C(d, ρ, Ω, α) such that +(2.17) +|∇xρδ,α(x, y)| ≤ C +� +ρηδ(x),α+1(|y − x|) + ρηδ(x),α+1(|y − x|) + ρηδ(y),α+1(|y − x|) +� +for all x, y ∈ Ω. +Proof. This follows by direct computation and the properties of ρ: +∇xρδ,α(x, y) = 1 +2 +�ρηδ(x),α(|y − x|) +ηδ(x)2 ++ +ρηδ(y),α(|y − x|) +ηδ(y)2 +� +(y − x) +− |y − x|2 +2ηδ(x)3 ρηδ(x),α(|y − x|)∇ηδ(x) − d + α +2ηδ(x)ρηδ(x),α(|y − x|)∇ηδ(x) . +Thus, using the support of ρ, we see the result. +□ + +NONLOCAL BOUNDARY-VALUE PROBLEMS +13 +Since ρ and ρ satisfy the hypotheses for the upper bound of Lemma 2.3, we have the +following corollary: +Corollary 2.7. Let Ω ⊂ Rd satisfy (AΩ), ρ satisfy (A1), α ≥ 0 and β ∈ [0, α + 1]. Then there +exists C = C(d, ρ, Ω, α, β) such that +(2.18) +ˆ +Ω +|ηδ(y)|β|∇xρδ,α(x, y)| dy ≤ +C +ηδ(x)1+α−β +for all x ∈ Ω. +Proof. We first apply (2.17) then repeatedly use (2.2)-(2.3) and (2.10). +□ +As a special case of (2.18) we have the estimate for the derivative of Φδ,α: +Corollary 2.8. Let Ω ⊂ Rd satisfy (AΩ), ρ satisfy (A1), α ≥ 0 and β ∈ [0, α + 1]. Then there +exists a constant C depending only on d, ρ, and Ω such that +(2.19) +|∇Φδ,α(x)| ≤ +ˆ +Ω +|∇xρδ,α(x, y)| dy ≤ +C +ηδ(x)1+α +∀ x ∈ Ω. +3. The nonlocal operator and the nonlocal Green’s identity +We first present some estimates on the nonlocal operator which helps us establish the +nonlocal Green’s identity. In turn, the identity allows us to further interpret the results of the +nonlocal operator acting on more general functions. +3.1. The operator on Lebesgue and Sobolev spaces. +Theorem 3.1. Let Ω ⊂ Rd satisfy (AΩ), ρ satisfy (A1), and p ∈ [1, ∞]. Then Lδu(x) defines a +measurable function on Ω for any u ∈ Lp(Ω). Moreover, with Φδ,2 defined as in (1.13), we have +Lδu(x) +Φδ,2(x) ∈ Lp(Ω) and there exists a constant C depending only on d, p and Ω such that +���� +Lδu +Φδ,2 +���� +Lp(Ω) +≤ C ∥u∥Lp(Ω) . +Proof. By Hölder’s inequality, +���� +Lδu +Φδ,2 +���� +p +Lp(Ω) += +ˆ +Ω +2p +Φδ,2(x)p +���� +ˆ +Ω +ρδ,2(x, y)(u(x) − u(y)) dy +���� +p +dx +≤ +ˆ +Ω +2p +Φδ,2(x)p +�ˆ +Ω +ρδ,2(x, y) dy +�p/p′ �ˆ +Ω +ρδ,2(x, y)|u(x) − u(y)|p dy +� +dx += +ˆ +Ω +2 +Φδ,2(x) +ˆ +Ω +ρδ,2(x, y)|u(x) − u(y)|p dy dx . +Then by (2.16) and Lemma 2.1 +ρδ,2(x, y) +Φδ,2(x) +≤ +1 +κ2,ℓ + + +ρ +� +|y−x| +ηδ(x) +� +ηδ(x)d ++ +ρ +� +|y−x| +ηδ(y) +� +ηδ(y)d +ηδ(x)2 +ηδ(y)2 + + ≤ C + + +ρ +� +|y−x| +ηδ(x) +� +ηδ(x)d ++ +ρ +� +|y−x| +ηδ(y) +� +ηδ(y)d + + + +14 +JAMES M. SCOTT AND QIANG DU +for all x y ∈ Ω, so +���� +Lδu +Φδ,2 +���� +p +Lp(Ω) +≤ C +ˆ +Ω +ˆ +Ω +� +ρηδ(x)(|y − x|) + ρηδ(y)(|y − x|) +� +|u(x) − u(y)|p dy dx +≤ C +ˆ +Ω +ˆ +Ω +ρηδ(x) (|y − x|) (|u(x)|p + |u(y)|p) dy dx ++ C +ˆ +Ω +ˆ +Ω +ρηδ(y)(|y − x|)(|u(x)|p + |u(y)|p) dy dx . +Then by Lemma 2.3 we have +���� +Lδu +Φδ,2 +���� +Lp(Ω) +≤ C(Ω, p) ∥u∥Lp(Ω) +as desired. For p = ∞ and x ∈ Ω, +���� +Lδu(x) +Φδ,2(x) +���� = |u(x) − Kδ,2u(x)| ≤ 2 ∥u∥L∞(Ω) . +□ +By Theorem 3.3, for any u ∈ Lp(Ω), Lδu(x) coincides with a Lebesgue-measurable function +Φδ,2(x)w(x) on Ω, where w ∈ Lp(Ω). Since Φδ,2(x) ≈ +1 +ηδ(x)2 , Lδu(x) clearly belongs to Lp +loc(Ω), +but it may not be globally integrable. +If we assume that u is smooth we can get a better result. To do this, we first present the +following pointwise characterization. +Lemma 3.2. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). Suppose that u ∈ C2(Ω).Then for +all δ ∈ (0, δ0) the quantity Lδu(x) defines a function in L∞(Ω). More precisely +(3.1) +Lδu(x) = 2F1(x) · ∇u(x) + F2(x, u) , +where F1 ∈ L∞(Ω; Rd) is defined by +(3.2) +F1(x) := +ˆ +Ω +ρδ,2(x, y)(y − x) dy +and satisfies +(3.3) +sup +x∈Ω +|F1(x)| ≤ Cχ{dist(x,∂Ω)≤ ¯C +√ +δ} , +for C(ρ, Ω) > 0 and ¯C(ρ, Ω) > 0 , +and F2 ∈ Lp(Ω) depends on ρ, Ω, δ and ∇2u, and satisfies +(3.4) +∥F2(·, u)∥Lp(Ω) ≤ C(ρ, Ω) +��∇2u +�� +Lp(Ω) +for any p ∈ [1, ∞] . +Proof. Taylor expansion gives +−Lδu(x) = 2 +ˆ +Ω +ρδ,2(x, y)(y − x) dy · ∇u(x) ++ 2 +ˆ +Ω +ρδ,2(x, y) +ˆ 1 +0 +� +∇2u(y + t(x − y))(x − y), (x − y) +� +(1 − t) dt dy . +By Lemma 2.3 applied to the kernel r2ρ(r) and α = 0, the Lp(Ω) norm of the last term (which +defines F2) is bounded by C(ρ, Ω) +��∇2u +�� +Lp(Ω). Further, since the set {y : |x − y| < R0ηδ(x)} +is compactly contained in Ω for all x ∈ Ω, +ˆ +Ω +ρηδ(x),2(|y − x|)(y − x) dy = +1 +ηδ(x) +ˆ +B(0,R0) +zρ(|z|) dz = 0, +∀x ∈ Ω . +So we just need to show that +(3.5) +|F1(x)| = +���� +ˆ +Ω +ρηδ(y),2(|y − x|)(y − x) dy +���� ≤ C(ρ, d, Ω)χ{dist(x,∂Ω)≤ ¯C +√ +δ}, +∀x ∈ Ω . + +NONLOCAL BOUNDARY-VALUE PROBLEMS +15 +Fix x ∈ Ω. +Note that the integral is absolutely convergent and bounded by Cηδ(x)−1 by +Lemma 2.3 applied to the kernel rρ(r) and α = 1. We will use a coordinate change. Define +gx(y) := y − x +ηδ(y) . +Then +∇gx(y) = +1 +ηδ(y)I − (y − x) ⊗ ∇ηδ(y) +ηδ(y)2 = +1 +ηδ(y) +� +I − (y − x) +ηδ(y) +⊗ ∇ηδ(y) +� +. +Now, recalling the definition of δ0 in (Aη), gx(y) is one-to-one on the boundary of the set +{y : |y − x| ≤ +R0 +1−κ1R0 +√ +δηδ(x)}, which for δ < δ0 contains {y : |y − x| ≤ R0ηδ(y)} by (2.3) and +is contained in Ω. Additionally, we use (2.2) and that |∇ηδ(y)| ≤ κ1 +√ +δ to obtain +ηδ(y) + ∇ηδ(y) · (x − y) ≥ ηδ(y) − κ1 +√ +δ|y − x| ≥ +� +1 − +2κ1 +√ +δR0 +1 − κ1 +√ +δR0 +� +ηδ(x) > 0 +for all y ∈ Ω satisfying |y − x| ≤ +R0 +1−κ1R0 +√ +δηδ(x) (which again holds assuming that δ < δ0). +Therefore on this set, +det ∇gx(y) = ηδ(y) + ∇ηδ(y) · (x − y) +ηδ(y)d+1 +is bounded away from zero independent of y. It follows that gx(y) is globally invertible on +{y : |y − x| ≤ R0ηδ(y)} and so +F1(x) = +ˆ +Ω +ρηδ(y),2(|y − x|)(y − x) dy += +1 +ηδ(x) +ˆ +{|x−y| 0 for all x ∈ Ω +it follows that Lδun(x) converges to Lδu(x) as n → ∞ for almost every x ∈ Ω. Therefore by +Fatou’s lemma and Lemma 3.2 +∥Lδu∥Lp(Ω) ≤ lim inf +n→∞ ∥Lδun∥Lp(Ω) ≤ lim inf +n→∞ +� +∥F1∇un∥Lp(Ω) + ∥F2(·, un)∥Lp(Ω) +� +≤ C(ρ, Ω) lim inf +n→∞ +� +∥∇un∥Lp(Ω) + +��∇2un +�� +Lp(Ω) +� +≤ C ∥u∥W 2,p(Ω) . +□ +3.2. The operator on the nonlocal function space. Concerning the action of the nonlocal +operator on functions in the nonlocal energy space, we have +Theorem 3.4. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). Let u ∈ Wδ,2(Ω). Then +Lδu(x) +√ +Φδ,2(x) ∈ +L2(Ω), and there exists constant C depending only on ρ and Ω such that +����� +Lδu +�Φδ,2 +����� +L2(Ω) +≤ C[u]Wδ,2(Ω) . +Proof. The conclusion follows from an application of Hölder’s inequality and the definition of +Φδ,2: +����� +Lδu +�Φδ,2 +����� +2 +L2(Ω) += +ˆ +Ω +22 +Φδ,2(x) +���� +ˆ +Ω +ρδ,2(x, y)(u(x) − u(y)) dy +���� +2 +dx +≤ +ˆ +Ω +22 +Φδ,2(x) +�ˆ +Ω +ρδ,2(x, y) dy +� �ˆ +Ω +ρδ,2(x, y)|u(x) − u(y)|2 dy +� +dx += 4 +ˆ +Ω +ˆ +Ω +ρδ,2(x, y)|u(x) − u(y)|2 dy dx ≤ C[u]2 +Wδ,2(Ω) . +□ +3.3. Nonlocal Green’s identity for smooth functions. +Theorem 3.5. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1)-(A2). Suppose u, v ∈ C2(Ω). Then +(1.5) holds. That is, +Bρ,δ(u, v) = +ˆ +Ω +Lδu(x) · v(x) dx + +ˆ +∂Ω +∂u +∂ν (x) · v(x) dσ(x) . +We begin with two lemmas. + +NONLOCAL BOUNDARY-VALUE PROBLEMS +17 +Lemma 3.6. Let Ω ⊂ Rd satisfy (AΩ). Suppose ρ ∈ C0([0, ∞)) with support in [0, R0] for fixed +R0 ∈ (0, 1). Then there exist constants C = C(ρ, R0, ¯κ0, δ) > 0 and b = b(R0, ¯κ0, δ) > 1 for +which the following holds: for any ε > 0 with ε ≪ δ +ˆ +Ω\Ω√ε +ρηδ(y),2(|y − x|) dy = +ˆ +Ω√ +bε\Ω√ε +ρηδ(y),2(|y − x|) dy ≤ Cε−2χΩ√ε\Ω√ +ε/b(x) +for all x ∈ Ω√ε. +Proof. Note that by Lemma 2.3 the integral is finite for any fixed x ∈ Ω. +Set b := +1 +(1−¯κ0R0 +√ +δ)2 . First, for ε ≪ δ we apply (2.5) with r = +√ +bε to see that {y : |y−x| ≤ +R0ηδ(y)} ⊂ Ω√ +bε whenever dist(x, ∂Ω) < √ε. This gives the first equality. +Second, for ε ≪ δ we again apply (2.5) but with r = √ε to see that, whenever dist(x, ∂Ω) < +� ε +b, we have {y : |y −x| ≤ R0ηδ(y)} ⊂ Ω√ε. Therefore thanks to the support of ρ, the integral +is zero if x ∈ Ω√ +ε/b. Thus by Lemma 2.3 +ˆ +Ω\Ω√ε +ρηδ(y),2(|y − x|) dy ≤ +CχΩ√ε\Ω√ +ε/b(x) +ηδ(x)2 +≤ +CχΩ√ε\Ω√ +ε/b(x) +ε2 +. +□ +Lemma 3.7. Let Ω ⊂ Rd satisfy (AΩ). Suppose ρ ∈ C0([0, ∞)) with support in [0, R0] for fixed +R0 ∈ (0, 1). Then there exist constants C = C(ρ, R0, ¯κ0, δ) > 0 and a = a(R0, ¯κ0, δ) > 1 for +which the following holds: for any ε > 0 with ε ≪ δ +ˆ +Ω\Ω√ε +ρηδ(x),2(|y − x|) dy = +ˆ +Ω√aε\Ω√ε +ρηδ(x),2(|y − x|) dy ≤ Cε−2χΩ√ε\Ω√ +ε/a(x) +for all x ∈ Ω√ε. +Proof. Note that by Lemma 2.3 the integral is finite for any fixed x ∈ Ω. +Set a := (1+ ¯κ0R0 +√ +δ)2. First, for ε ≪ δ we apply (2.4) with r = √aε to see that, whenever +dist(x, ∂Ω) < √ε, {y : |y − x| ≤ R0ηδ(x)} ⊂ Ω√aε. This gives the first equality. +Second, for ε ≪ δ we again apply (2.4) but with r = √ε to see that {y : |y − x| ≤ +R0ηδ(x)} ⊂ Ω√ε whenever dist(x, ∂Ω) < � ε +a. Therefore thanks to the support of ρ the integral +is zero if x ∈ Ω√ +ε/a. Thus by Lemma 2.3 +ˆ +Ω\Ω√ε +ρηδ(x),2(|y − x|) dy ≤ +CχΩ√ε\Ω√ +ε/a(x) +ηδ(x)2 +≤ +CχΩ√ε\Ω√ +ε/a(x) +ε2 +. +□ +Proof of Theorem 3.5. Let ε > 0 with ε ≪ δ. Since +ρδ,2(x, y)|u(x) − u(y)| |v(x) − v(y)| ≤ ∥∇u∥L∞(Ω) ∥∇v∥L∞(Ω) ρδ,2(x, y)|x − y|2 ∈ L1(Ω × Ω) , +by the dominated convergence theorem +Bρ,δ(u, v) = +ˆ +Ω +ˆ +Ω +ρδ,2(x, y)(u(x) − u(y))(v(x) − v(y)) dy dx += lim +ε→0 +ˆ +Ω\Ω√ε +ˆ +Ω\Ω√ε +ρδ,2(x, y)(u(x) − u(y))(v(x) − v(y)) dy dx . + +18 +JAMES M. SCOTT AND QIANG DU +Now we can use nonlocal integration by parts on the truncated form as well as linearity of the +integrals thanks to their absolute convergence guaranteed by Lemma 3.6 and Lemma 3.2; we get +Bρ,δ(u, v) = +ˆ +Ω\Ω√ε +ˆ +Ω\Ω√ε +ρδ,2(x, y)(u(x) − u(y))(v(x) − v(y)) dy dx += 2 +ˆ +Ω\Ω√ε +ˆ +Ω\Ω√ε +ρδ,2(x, y)(u(x) − u(y)) dy v(x) dx += −2 +ˆ +Ω\Ω√ε +ˆ +Ω√ε +ρδ,2(x, y)(u(x) − u(y))v(x) dy dx + +ˆ +Ω\Ω√ε +Lδu(x)v(x) dx += −2 +ˆ +Ω\Ω√ε +ˆ +Ω√ε +ρδ,2(x, y)(u(x) − u(y))v(y) dy dx +− 2 +ˆ +Ω\Ω√ε +ˆ +Ω√ε +ρδ,2(x, y)(u(x) − u(y))(v(x) − v(y)) dy dx + +ˆ +Ω\Ω√ε +Lδu(x)v(x) dx . +The dominated convergence theorem gives that the second integral vanishes and that +lim +ε→0 +ˆ +Ω\Ω√ε +Lδu(x)v(x) dx = +ˆ +Ω +Lδu(x)v(x) dx , +since |Lδu(x)v(x)| ≤ C(ρ, d, Ω) ∥u∥W 2,∞(Ω) ∥v∥L∞(Ω) < ∞. Therefore it remains to show that +(3.9) +lim +ε→0 +ˆ +Ω√ε +ˆ +Ω\Ω√ε +2 ρδ,2(x, y)(u(x) − u(y))v(x) dy dx = +ˆ +∂Ω +∂u +∂ν (x)v(x) dx . +Now on the left-hand side integrand we perform a second-order Taylor expansion of u(y) at +x and estimate the remainder term involving ∇2u. Using the arguments of Lemma 3.6 and +Lemma 3.7 we see that +ˆ +Ω√ε +ˆ +Ω\Ω√ε +ρδ,2(x, y)|x − y|2 dy dx ≤ C(Ω)|Ω√ε| = o(1) , +so (3.9) is equivalent to showing that +(3.10) +lim +ε→0 +ˆ +Ω√ε +ˆ +Ω\Ω√ε +2 ρδ,2(x, y)(x − y) · ∇u(x)v(x) dy dx = +ˆ +∂Ω +∂u +∂ν (x)v(x) dx . +Step 1: Localization. For any x0 ∈ ∂Ω, denote the outward and inward unit normal vectors +as ν(x0) and n(x0), respectively. +Since Ω is a C2 domain, it satisfies the uniform interior- +exterior sphere condition. As a consequence, there exists ε0 > 0 depending only on Ω such that +the following holds: for any x0 ∈ ∂Ω there exists a Euclidean ball B = B(x0, β) ⊂ Rd such that +the map Ψ : ∂Ω ∩ B × [0, ε0) → Ω2ε0 ∩ B(x0, 2β) given by +Ψ(x′, t) := x′ + tn(x′) +is a diffeomorphism onto its image. Furthermore, (choosing ε0 smaller if necessary) for every x ∈ +Ωε0 there exists a unique nearest-point projection ξ(x) ∈ ∂Ω such that |x − ξ(x)| = dist(x, ∂Ω). +We collect the relevant computations concerning the distance function and the map Ψ: +dist(x′ + tn(x′), ∂Ω) = t, +∀(x′, t) ∈ ∂Ω ∩ B × [0, ε0) , +ξ(x′ + tn(x′)) = x′, +∀(x′, t) ∈ ∂Ω ∩ B × [0, ε0) , +∇ dist(x′ + tn(x′), ∂Ω) = n(x′), +∀(x′, t) ∈ ∂Ω ∩ B × [0, ε0) . +(3.11) +See [38] for further details. It is clear upon parametrizing ∂Ω (choosing the radius β smaller if +necessary so that ∂Ω ∩ B can be written as the graph of a function) that +(3.12) +det ∇Ψ(x′, 0)dH d−1(x′) = dσ(x′) on ∂Ω ∩ B , +where H d−1 denotes d−1-dimensional Hausdorff measure. Hereafter we abbreviate dH d−1(x′) +as dx′ whenever it is clear from context. + +NONLOCAL BOUNDARY-VALUE PROBLEMS +19 +Now, there exists a finite collection of Euclidean balls {Bi}N +i=1, Bi = B((x0)i, βi) whose +centers are points on ∂Ω that cover ∂Ω and such that (3.11)-(3.12) are satisfied for ∂Ω ∩ Bi. +Therefore by a localization argument with a partition of unity subordinate to the cover {Bi}N +i=1 +it suffices to find the limit (3.10) for ∇u and v compactly supported on Ω ∩ Bi for some i. In +what follows we suppress the index to simplify notation. +Step 2: Let ε ≪ min{ε0, δ}. We show that +lim +ε→0 +ˆ +Ω√ε +ˆ +Ω\Ω√ε +ρηδ(x),2(|x − y|)(y − x) · ∇u(x)v(x) dy dx += − +ˆ +B(0,R0)∩{zd>0} +z2 +dρ (|z|) dz · +ˆ +∂Ω +∂u +∂ν (x)v(x) dσ(x) , +(3.13) +where u and v are supported on ∂Ω ∩ B where B is as in Step 1. To begin, we write the outer +integral using the change of variables x = Ψ(x′, t) as well as using (3.11) and Lemma 3.7: +ˆ +Ω√ε +ˆ +Ω\Ω√ε +ρηδ(x),2(|x − y|)(y − x) · ∇u(x)v(x) dy dx += +ˆ +Ω√ε +ˆ +Rd χΩ√aε\Ω√ε(y + x) ρηδ(x),2(|y|)y · ∇u(x)v(x) dy dx += +ˆ +∂Ω∩B +ˆ √ε +0 +ˆ +Rd +� +χΩ√aε\Ω√ε(Ψ(x′, t) + y) +1 +t2d+4 ρ +�|y| +t2 +� +y · ∇u(Ψ(x′, t))v(Ψ(x′, t)) det ∇Ψ(x′, t) +� +dy dt dx′ += +ˆ +∂Ω∩B +ˆ √ε +0 +ˆ +Rd +� +χΩ√aε\Ω√ε(Ψ(x′, t) + t2y) 1 +t2 ρ (|y|) +y · ∇u(Ψ(x′, t))v(Ψ(x′, t)) det ∇Ψ(x′, t) +� +dy dt dx′ . +Now, fix x′ ∈ ∂Ω ∩ B. We define ⟨n(x′)⟩⊥ := {y′ ∈ Rd : y′ · n(x′) = 0}. Use the change +of variables y = y′ + τn(x′) for y′ ∈ ⟨n(x′)⟩⊥ in the inner integral as well as the fact that +supp ρ ⊂ [0, R0] to get +ˆ +Ω√ε +ˆ +Ω\Ω√ε +ρηδ(x),2(|x − y|)(y − x) · ∇u(x)v(x) dy dx += +ˆ +∂Ω∩B +ˆ √ε +0 +ˆ +⟨n(x′)⟩⊥ +ˆ +R +� +χΩ√aε\Ω√ε(Ψ(x′, t) + t2(y′ + τn(x′))) χB(0,R0)(y′ + τn(x′)) +ρ (|y′ + τn(x′)|) +t2 +� +y′ + τn(x′) +� +· ∇u(Ψ(x′, t))v(Ψ(x′, t)) det ∇Ψ(x′, t) +� +dτ dy′ dt dx′ . +Now we analyze the indicator functions. Taylor-expanding ηδ(x) = (dist(x, ∂Ω))2 at the +point Ψ(x′, t) = x′ + tn(x′) gives +ηδ(Ψ(x′, t) + t2(y′ + τn(x′))) = t2 + 2tn(x′) · (t2(y′ + τn(x′))) + Π(t) += t2(1 + 2tτ) + Π(t) . +The function Π(t) depends on the Hessian of (dist(x, ∂Ω))2, and is also a function of x′ ∈ ∂Ω, +y′ ∈ ⟨n(x′)⟩⊥ and τ ∈ R. However, since Ω is C2 it is continuous and uniformly bounded with +respect to these variables and with respect to ε ∈ (0, ε0), and satisfies +1 +M t4 ≤ Π(t) ≤ Mt4 , + +20 +JAMES M. SCOTT AND QIANG DU +see [38]. Therefore +ˆ +Ω√ε +ˆ +Ω\Ω√ε +ρηδ(x),2(|x − y|)(y − x) · ∇u(x)v(x) dy dx += +ˆ +∂Ω∩B +ˆ √ε +0 +ˆ +⟨n(x′)⟩⊥ +ˆ +R +{ χ{ε 0 sufficiently small 0 ≤ �h−1 +ε,τ(1) ≤ 1 ≤ �h−1 +ε,τ(a) for all +t ∈ (0, 1) and for all τ ∈ (−R0, R0). It is also clear that on the region of integration the implicit +relation τ ≥ Π(√ε) +2ε +is satisfied. Further, the Lipschitz continuity of ∇u, v and ∇Ψ means that +the limit as ε → 0 will be unchanged if we replace √εt in their arguments with 0. Combining +all of this, we use Fubini’s theorem to get +ˆ +∂Ω∩B +ˆ +⟨n(x′)⟩⊥ +ˆ 1 +0 +ˆ +R +{ χ{1<�hε,τ (t) 0 sufficiently small, with +bounds depending only on R0. Thus +���� +1 +A − 1 +B +���� = +1 +|A||B| |A − B| ≤ C|hε,τ(A) − hε,τ(B)| = |1 − Π(√εA) +ε +− 1| ≤ Cε . +So we have +ˆ +∂Ω∩B +ˆ +⟨n(x′)⟩⊥ +ˆ R0 +Π(√ε) +2ε +{ 1 +√ε +� +1 +�h−1 +ε,τ(1) +− 1 +� +χB(0,R0)(y′ + τn(x′))ρ +� +|y′ + τn(x′)| +� +� +y′ + τn(x′) +� +· ∇u(x′)v(x′) det ∇Ψ(x′, 0) +� +dτ dy′ dx′ += +ˆ +∂Ω∩B +ˆ +⟨n(x′)⟩⊥ +ˆ R0 +Π(√ε) +2ε +{ 1 +√ε +� +1 +h−1 +ε,τ(1) − 1 +� +χB(0,R0)(y′ + τn(x′))ρ +� +|y′ + τn(x′)| +� +� +y′ + τn(x′) +� +· ∇u(x′)v(x′) det ∇Ψ(x′, 0) +� +dτ dy′ dx′ + O(√ε) . + +NONLOCAL BOUNDARY-VALUE PROBLEMS +21 +Thanks to the explicit formula for hε,τ(t), we can find the unique real root of the function +hε,τ(t) − 1 in terms of ε and τ. Then a direct computation gives +lim +ε→0 +1 +√ε +� +1 +h−1 +ε,τ(1) − 1 +� += τ . +By the dominated convergence theorem we can conclude the equality of limits +lim +ε→0 +ˆ +Ω√ε +ˆ +Ω\Ω√ε +ρηδ(x),2(|x − y|)(y − x) · ∇u(x)v(x) dy dx += lim +ε→0 +ˆ +∂Ω∩B +ˆ +⟨n(x′)⟩⊥ +ˆ R0 +Π(√ε) +2ε +� 1 +√ε +� +1 +h−1 +ε,τ(1) − 1 +� +χB(0,R0)(y′ + τn(x′)) +ρ +� +|y′ + τn(x′)| +� � +y′ + τn(x′) +� +· ∇u(x′)v(x′) det ∇Ψ(x′, 0) +� +dτ dy′ dx′ += +ˆ +∂Ω∩B +ˆ +⟨n(x′)⟩⊥ +ˆ R0 +0 +{ χB(0,R0)(y′ + τn(x′))τ ρ +� +|y′ + τn(x′)| +� +� +y′ + τn(x′) +� +· ∇u(x′)v(x′) det ∇Ψ(x′, 0) +� +dτ dy′ dx′ . +We can see that ˆ +∂Ω∩B +ˆ +⟨n(x′)⟩⊥ +ˆ R0 +0 +� +χB(0,R0)(y′ + τn(x′))τ ρ +� +|y′ + τn(x′)| +� +y′ · ∇u(x′)v(x′) det ∇Ψ(x′, 0) +� +dτ dy′ dx′ = 0 , +since the region of integration is radially symmetric with respect to y′. Further, since (y′ + +τn(x′)) · n(x′) = τ and {y′ + τn(x′) : y′ · n(x′) = 0 , τ > 0 , |y′|2 + τ 2 ≤ R2 +0} = B(0, R0) ∩ {z : +z · n(x′) > 0}, we can rewrite the τ-y′ integral to get +lim +ε→0 +ˆ +Ω√ε +ˆ +Ω\Ω√ε +ρηδ(x),2(|x − y|)(y − x) · ∇u(x)v(x) dy dx += +ˆ +∂Ω∩B +ˆ +⟨n(x′)⟩⊥ +ˆ R0 +0 +� +χB(0,R0)(y′ + τn(x′))τ 2 ρ +� +|y′ + τn(x′)| +� +n(x′) · ∇u(x′)v(x′) det ∇Ψ(x′, 0) +� +dτ dy′ dx′ += +ˆ +∂Ω∩B +ˆ +B(0,R0)∩{z·n(x′)>0} +� +|z · n(x′)|2ρ (|z|) n(x′) · ∇u(x′)v(x′) det ∇Ψ(x′, 0) +� +dz dx . +Applying the appropriate coordinate rotation in the z integral and noting the definition of surface +measure from Step 1, we obtain (3.13). +Step 3: Let ε ≪ min{ε0, δ}. We show that +lim +ε→0 +ˆ +Ω√ε +ˆ +Ω\Ω√ε +ρηδ(y),2(|x − y|)(y − x) · ∇u(x)v(x) dy dx += − +ˆ +B(0,R0)∩{zd>0} +z2 +dρ (|z|) dz · +ˆ +∂Ω +∂u +∂ν (x)v(x) dσ(x) , +(3.14) +where u and v are supported on ∂Ω ∩ B where B is as in Step 1. +We proceed in a similar way. Since ε ≪ δ, we have that ηδ(y) = dist(y, ∂Ω)2 on Ω√ +bε \Ω√ε. +Now by Lemma 3.6 we can use linearity of the integrals to write +ˆ +Ω\Ω√ε +ρηδ(y),2(|y − x|)(y − x) dy += +ˆ +Ω√ +bε\Ω√ε +ρ +�|y − x| +ηδ(y) +� (y − x) +ηδ(x) +ηδ(y) + ∇ηδ(y) · (x − y) +ηδ(y)d+2 +dy ++ +ˆ +Ω√ +bε\Ω√ε +ρ +�|y − x| +ηδ(y) +� (y − x) +ηδ(x) +ηδ(x) − ηδ(y) − ∇ηδ(y) · (x − y) +ηδ(y)d+2 +dy . + +22 +JAMES M. SCOTT AND QIANG DU +Taking the inner product with v∇u and integrating with respect to x ∈ Ω√ε, we see that by an +argument identical to the proof of Lemma 3.2 the second integral is majorized by +C(d, ρ, Ω) ∥∇u∥L∞(Ω) ∥v∥L∞(Ω) |Ω√ε| , +and thus tends to zero as ε → 0. Therefore +lim +ε→0 +ˆ +Ω√ε +ˆ +Ω\Ω√ε +ρηδ(y),2(|x − y|)(y − x) · ∇u(x)v(x) dy dx += lim +ε→0 +ˆ +Ω√ε +ˆ +Ω√ +bε\Ω√ε +ρ +�|y − x| +ηδ(y) +� (y − x) +ηδ(x) +· ∇u(x)v(x)ηδ(y) + ∇ηδ(y) · (x − y) +ηδ(y)d+2 +dy dx . +(3.15) +We recall the definition gx(y) = y−x +ηδ(y), and that det ∇gx(y) = ηδ(y)+∇ηδ(y)·(x−y) +ηδ(y)d+1 +. Thus +ˆ +Ω√ε +ˆ +Ω√ +bε\Ω√ε +ρ +�|y − x| +ηδ(y) +� (y − x) +ηδ(x) · ∇u(x)v(x)ηδ(y) + ∇ηδ(y) · (x − y) +ηδ(y)d+2 +dy dx += +ˆ +Ω√ε +ˆ +Rd χΩ√ +bε\Ω√ε(g−1 +x (z)) +1 +ηδ(x)ρ (|z|) z · ∇u(x)v(x) dz dx . +(3.16) +Now we Taylor expand the distance function; since the integration region is ε ≤ ηδ(g−1 +x (z)) ≤ bε +and by definition of gx we have +ηδ(g−1 +x (z)) = ηδ(x) + ∇ηδ(x) · (g−1 +x (z) − x) + O(|g−1 +x (z) − x|2) += ηδ(x) + ηδ(g−1 +x (z))∇ηδ(x) · z + O(|z|2ηδ(g−1 +x (z))2) , +which implies +(3.17) +ηδ(g−1 +x (z)) = +ηδ(x) +1 − ∇ηδ(x) · z + O(ε2) . +We perform the two changes of variables just as in Step 2: +x = Ψ(x′, t) = x′ + tn(x′) , +z = z′ + τn(x′) , z′ ∈ +� +n(x′) +�⊥ . +Therefore by (3.11) and (3.17), we get +ˆ +Ω√ε +ˆ +Rd χΩ√ +bε\Ω√ε(g−1 +x (z)) +1 +ηδ(x)ρ (|z|) z · ∇u(x)v(x) dz dx += +ˆ +Ω∩B +ˆ √ε +0 +ˆ +⟨n(x′)⟩⊥ +ˆ +R +χ{ε< +t2 +1−2δtτ +O(ε2)0} +z2 +dρ (|z|) dz · +ˆ +∂Ω +∂u +∂ν (x)v(x) dσ(x) . + +NONLOCAL BOUNDARY-VALUE PROBLEMS +23 +Combining this with (3.16) and (3.15) completes Step 3, and combining (3.13)-(3.14) gives +lim +ε→0 +ˆ +Ω√ε +ˆ +Ω\Ω√ε +2 ρδ,2(x, y)(x − y) · ∇u(x)v(x) dy dx += 2 +ˆ +B(0,R0)∩{zd>0} +z2 +dρ (|z|) dz · +ˆ +∂Ω +∂u +∂ν (x)v(x) dσ(x) . +By (A2) the first integral is equal to 1, and so (3.10) is proved. +□ +3.4. Interpreting the action of the operator as a distribution. With the help of the +nonlocal Green’s identity, we can interpret the action of the nonlocal operator on Sobolev spaces +and the nonlocal energy space in the sense of distributions, as demonstrated in the following +theorems. Such interpretations are very useful in the proofs of the well-posedness of nonlocal +boundary value problems. +Theorem 3.8. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). +Let u ∈ H1(Ω). +Define the +distribution Lδu ∈ D′(Ω) by +(3.18) +⟨Lδu, v⟩ := lim +n→∞ +ˆ +Ω +Lδun(x)v(x) dx , +v ∈ C∞ +c (Ω) , +where {un} is a sequence of functions in C2(Ω) that converges to u in H1(Ω). Then the definition +of Lδu is independent of the sequence chosen, and the action of Lδu can be written explicitly as +(3.19) +⟨Lδu, v⟩ = Bρ,δ(u, v) , +v ∈ C∞ +c (Ω) . +Furthermore, Lδu ∈ H−1(Ω), there exists C = C(ρ, Ω) such that +∥Lδu∥H−1(Ω) ≤ C ∥u∥H1(Ω) , +and the action of Lδu can be written as (3.19) for any v ∈ H1 +0(Ω). +Proof. For each n the integral defining ⟨Lδun, v⟩ is absolutely convergent thanks to Theorem 3.3, +and from the nonlocal Green’s identity for smooth functions established in Theorem 3.5, (1.17), +and Theorem 1.5 we obtain +���� +ˆ +Ω +Lδun(x)v(x) dx +���� = |Bρ,δ(u, v)| +≤ C(ρ, Ω) ∥un∥H1(Ω) ∥v∥H1(Ω) . +Each conclusion in the theorem then follows from this estimate. +□ +A similar result holds when considering the action on functions in the nonlocal energy space. +Theorem 3.9. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). Let u ∈ Wδ,2(Ω), and define +the distribution Lδu ∈ D′(Ω) by (3.18), where {un} is a sequence of functions in C∞(Ω) that +converges to u in Wδ,2(Ω). Then the definition of Lδu is independent of the sequence chosen, +and the action of Lδu can be written explicitly as (3.19). Furthermore, Lδu ∈ [Wδ,2 +0 (Ω)]∗, there +exists C = C(ρ, Ω) such that +(3.20) +∥Lδu∥[Wδ,2 +0 (Ω)]∗ ≤ C ∥u∥Wδ,2(Ω) , +and the action of Lδu can be written as (3.19) for any v ∈ Wδ,2 +0 (Ω). +Proof. For each n the integral defining ⟨Lδun, v⟩ is absolutely convergent thanks to Theorem 3.3, +and from the nonlocal Green’s identity for smooth functions established in Theorem 3.5 and +(1.17) we obtain the estimate +���� +ˆ +Ω +Lδun(x)v(x) dx +���� = |Bρ,δ(u, v)| ≤ C(ρ, Ω) ∥un∥Wδ,2(Ω) ∥v∥Wδ,2(Ω) . +Each conclusion in the theorem then follows from this estimate. +□ + +24 +JAMES M. SCOTT AND QIANG DU +4. The nonlocal function space: trace theorem and nonlocal Poincaré +inequalities +In this section, we present a few important properties on the nonlocal energy spaces such +as the trace theorems and Poincare inequalities. The nonlocal Green’s identity in more general +function spaces is also established. All of these results are important ingredients in the later +proofs of the well-posedness of nonlocal problems with local boundary conditions. +4.1. The trace operator. Intuitively, we expect that the choice of the heterogenenous local- +ization function ηδ picked in the current work leads to a stronger localization effect on the +boundary than the case of localization function being linearly proportional to dist(x, ∂Ω). Since +the nonlocal energy space associated with the latter choice enjoys the same trace inequality as +W 1,p [26,34,58], we expect that the new nonlocal energy space associated with ηδ would share a +similar trace inequality. While this can indeed be shown rigorously, we only briefly present the +main results but will leave full details (in more general forms) to separate works. +Theorem 4.1. Let Ω ⊂ Rd satisfy (AΩ), and let R0 ∈ (0, 1). For all δ < δ0, there exists a +constant C depending on Ω, R0, and δ such that +ˆ +Ω +ˆ +{|y−x|≤R0 dist(x,∂Ω)} +|u(x) − u(y)|2 +dist(x, ∂Ω)d+2 dy dx ≤ C +� +∥u∥2 +L2(Ω) + [u]2 +Wδ,2(Ω) +� +. +It is a consequence of the above theorem that the properties of the nonlocal function spaces +studied in [26,58] are inherited by the space Wδ,2(Ω). For instance, we have a trace theorem. +Theorem 4.2. Let Ω ⊂ Rd satisfy (AΩ). Let T denote the trace operator, i.e. for u ∈ C∞(Ω) +Tu = u +�� +∂Ω . +Then for each δ < δ0 the trace operator extends to a bounded linear operator T : Wδ,2(Ω) → +H +1 +2(∂Ω), and there exists C = C(Ω, R0, δ) such that +∥Tu∥H +1 +2 (∂Ω) ≤ C ∥u∥Wδ,2(Ω) , +∀u ∈ Wδ,2(Ω) . +Proof. Follows from [58, Theorem 1.3] and Theorem 4.1. +□ +Corollary 4.3. Let Ω ⊂ Rd satisfy (AΩ). Suppose u ∈ Wδ,2 +0 (Ω). Then Tu = 0 . +The trace theorems ensure that proper local boundary conditions can be imposed for the +associated nonlocal problems. +4.2. The nonlocal Green’s identity for wider classes of functions. +Theorem 4.4. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). Let u ∈ Wδ,2(Ω) and suppose +additionally that Lδu ∈ L2(Ω). Then the operator +∂ +∂ν defined for functions v ∈ C∞(Ω) as +∂v +∂ν = ∇v(x) · ν(x) , +x ∈ ∂Ω , +extends to an action on the function u. To be precise, +∂ +∂ν maps u to ∂u +∂ν ∈ H− 1 +2 (∂Ω), with +���� +∂u +∂ν +���� +H− 1 +2 (Ω) +≤ C(Ω, ρ) +� +[u]Wδ,2(Ω) + ∥Lδu∥L2(Ω) +� +. +Proof. Define the Banach space +Xδ,2(Ω) := { closure of C∞(Ω) with respect to the norm ∥·∥Xδ,2(Ω)} , + +NONLOCAL BOUNDARY-VALUE PROBLEMS +25 +where the norm is defined as +∥u∥2 +Xδ,2(Ω) := ∥u∥2 +Wδ,2(Ω) + ∥Lδu∥2 +L2(Ω) . +It is clear that Xδ,2(Ω) is a closed subspace of Wδ,2(Ω). Let {un} be a sequence in C∞(Ω) that +converges to u in Xδ,2(Ω). Then by the nonlocal Green’s identity (1.5) already established in +Theorem 3.5 for smooth functions we have for any v ∈ C∞(Ω) +ˆ +∂Ω +∂un +∂ν (x) · v(x) dσ(x) = Bρ,δ(un, v) − +ˆ +Ω +Lδun(x) · v(x) dx . +By applying Hölder’s inequality to the right-hand side, we get +(4.1) +���� +ˆ +∂Ω +∂un +∂ν (x) · v(x) dσ(x) +���� ≤ C(Ω, ρ) +� +[un]Wδ,2(Ω) + ∥Lδun∥L2(Ω) +� +∥v∥Wδ,2(Ω) . +Now for ¯v ∈ H +1 +2 (∂Ω), let v ∈ H1(Ω) be any extension of ¯v to Ω with ∥v∥H1(Ω) ≤ C ∥¯v∥H +1 +2 (∂Ω), +where c is independent of ¯v and v. Let {vj} be a sequence in C∞(Ω) converging to v in H1(Ω). +Then for any n ∈ N we can use the dominated convergence theorem and the classical trace +theorem for H1(Ω) to conclude that +ˆ +∂Ω +∂un +∂ν (x) · ¯v(x) dσ(x) = lim +j→∞ +ˆ +∂Ω +∂un +∂ν (x) · Tvj(x) dσ(x) += lim +j→∞ Bρ,δ(un, vj) − lim +j→∞ +ˆ +Ω +Lδun(x) · vj(x) dx += Bρ,δ(un, v) − +ˆ +Ω +Lδun(x) · v(x) dx . +By (4.1) and by Theorem 1.5 +���� +ˆ +∂Ω +∂un +∂ν (x) · ¯v(x) dσ(x) +���� ≤ C +� +[un]Wδ,2(Ω) + ∥Lδun∥L2(Ω) +� +∥v∥Wδ,2(Ω) +≤ C(Ω, ρ) +� +[u]Wδ,2(Ω) + ∥Lδu∥L2(Ω) +� +∥¯v∥H +1 +2 (Ω) , +(4.2) +and so ∂un +∂ν defines an object in H− 1 +2(∂Ω). +Finally we define the functional ∂u +∂ν by taking n → ∞: +∂u +∂ν := lim +n→∞ +∂un +∂ν , +where the limit is taken in the H− 1 +2 (Ω)-norm. The estimate (4.2) shows that the definition of +∂u +∂ν is independent of the approximating sequence {un} ⊂ Xδ,2(Ω) chosen, and that +���� +∂u +∂ν +���� +H− 1 +2 (Ω) +≤ C(Ω, ρ) +� +[u]Wδ,2(Ω) + ∥Lδu∥L2(Ω) +� +. +□ +Proposition 4.5. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). Then the nonlocal Green’s +identity (1.5) holds in the following cases: +i) u ∈ Wδ,2(Ω), v ∈ C∞ +c (Ω), +ii) u ∈ Wδ,2(Ω) with Lδu ∈ L2(Ω), v ∈ H1(Ω). +Proof. We begin with case i). Let {un} be a sequence in C∞(Ω) converging to u in the Wδ,2(Ω) +norm. Then for each n we can apply the nonlocal Green’s identity (1.5) proved for smooth +functions in Theorem 3.5 to get +Bρ,δ(un, v) = +ˆ +Ω +Lδun(x) · v(x) dx , + +26 +JAMES M. SCOTT AND QIANG DU +since v vanishes near ∂Ω. Now, an application of Hölder’s inequality shows that +lim +n→∞Bρ,δ(un, v) = Bρ,δ(u, v) . +To complete the proof in case i) we need to show that +(4.3) +lim +n→∞ +ˆ +Ω +Lδun(x) · v(x) dx = +ˆ +Ω +Lδu(x) · v(x) dx . +By (2.16) and since dist(suppv, ∂Ω) > 0 +|(Lδun(x) − Lδu(x))v(x)| ≤ |Lδun(x) − Lδu(x)| +� +Φδ,2(x) +· +� +Φδ,2(x)|v(x)| +≤ |Lδun(x) − Lδu(x)| +� +Φδ,2(x) +· +√κ2,u +ηδ(x) |v(x)| +≤ +C(Ω, ρ) +dist(suppv, ∂Ω) +|Lδun(x) − Lδu(x)| +� +Φδ,2(x) +· |v(x)| . +Therefore (4.3) follows from Hölder’s inequality and Theorem 3.4. +For case ii), recall the definition of Xδ,2(Ω) from Theorem 4.4. +let {un} and {vm} be +sequences in C∞(Ω) with {un} converging to u in the Xδ,2(Ω) norm and {vm} converging to v +in the H1(Ω) norm. Then for each n and m we can apply the nonlocal Green’s identity (1.5) +proved in Theorem 3.5 to get +Bρ,δ(un, vm) = +ˆ +Ω +Lδun(x) · vm(x) dx + +ˆ +∂Ω +∂un +∂ν (x) · vm(x) dσ(x) . +An application of Hölder’s inequality and the convergence properties of {un} and {vm} +shows that +lim +n→∞ lim +m→∞Bρ,δ(un, vm) = Bρ,δ(u, v) +and +lim +n→∞ lim +m→∞ +ˆ +Ω +Lδun(x)vm(x) dx = +ˆ +Ω +Lδu(x)v(x) dx . +To complete the proof in case ii) we need to show that +(4.4) +lim +n→∞ lim +m→∞ +ˆ +∂Ω +∂un +∂ν (x) · vm(x) dσ(x) = +ˆ +∂Ω +∂u +∂ν (x) · v(x) dσ(x) . +But this follows thanks to the classical trace theorem for H1(Ω) and Theorem 4.4. +□ +Remark 4.6. In the classical case, the additional assumption ∆u ∈ L2(Ω) is nec ∂u +∂ν ∈ H−1/2(∂Ω) +for functions +4.3. Poincaré inequalities. We define the Hilbert space +Wδ,2 +0 (Ω) := { closure of C∞ +c (Ω) with respect to the norm ∥·∥Wδ,2(Ω)} . +The following nonlocal Poincar’e inequalities, designed for either Dirichlet or Neumann +problems respectively, are proved in Section 6.2. +Theorem 4.7. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). +Then there exists a constant +CD = CD(d, Ω, ρ) > 0 such that for all δ < δ0 and u ∈ Wδ,2 +0 (Ω), +∥u∥L2(Ω) ≤ CD[u]Wδ,2(Ω) . +Theorem 4.8. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1), and recall the definition of Φδ = Φδ,0 +in (1.13). Then there exists a constant CN(d, Ω, ρ) > 0 such that for all δ < δ0 and u ∈ Wδ,2(Ω), +∥u − (Φδu)Ω∥L2(Ω) ≤ CN[u]Wδ,2(Ω) . + +NONLOCAL BOUNDARY-VALUE PROBLEMS +27 +5. Boundary-value problems: well-posedness results +Equipped with the nonlocal Green’s identity, we can now state weak versions of boundary- +value problems associated to Lδ with different boundary conditions. In the case of Dirichlet +data, the well-posedness of the homogeneous problem is established, and then the general inho- +mogeneous problem is treated by using auxiliary functions to reduce to the homogeneous case. +In the case of Neumann data, we can treat the inhomogeneous problem directly. +5.1. The Dirichlet problem: homogeneous boundary conditions. The first boundary- +value problem we treat is a nonlocal Poisson problem with homogeneous Dirichlet boundary +conditions as stated in (1.6)-(1.10). +Definition 5.1. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). For f ∈ [Wδ,2 +0 (Ω)]∗, we say that +u ∈ Wδ,2 +0 (Ω) is a weak solution to the nonlocal problem (1.6) with homogeneous Dirichlet data +(1.10) if +(5.1) +Bρ,δ(u, v) = ⟨f, v⟩ , +∀v ∈ Wδ,2 +0 (Ω) . +Theorem 5.2 (Well-posedness). Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). Let δ < δ0. Then +there exists a constant C = C(d, ρ, Ω, CD) such that +(5.2) +∥u∥2 +Wδ,2(Ω) ≤ CBρ,δ(u, u), +∀u ∈ Wδ,2 +0 (Ω) . +Moreover, for any f ∈ [Wδ,2 +0 (Ω)]∗ there exists a unique solution u ∈ Wδ,2 +0 (Ω) to (5.1) satisfying +the energy estimate +(5.3) +∥u∥Wδ,2(Ω) ≤ C(d, ρ, Ω) ∥f∥[Wδ,2 +0 (Ω)]∗ . +Proof. The coercivity (5.2) follows from (1.16) and Theorem 4.7. We conclude the existence and +uniqueness of a weak solution via the Lax-Milgram theorem thanks to the continuity of Bρ,δ +established in (1.17). +□ +5.2. The Dirichlet problem: inhomogeneous boundary conditions. Consider the prob- +lem (1.6) with inhomogenenous Dirichlet boundary data (1.7). This problem can be reduced +to the case of homogeneous Dirichlet boundary conditions by the following argument. +Let +G ∈ H1(Ω) be an extension of g to Ω with ∥G∥H1(Ω) ≤ C ∥g∥H +1 +2 (∂Ω). Then can we define the +solution u of (1.6)-(1.7) as +(5.4) +u(x) := w(x) + G(x) , +where w(x) is the unique weak solution of +(5.5) +� +Lδw = f − LδG +in Ω , +w = 0 +on ∂Ω . +Theorem 5.3 (Well-posedness). Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). Let δ < δ0. Let +f ∈ [Wδ,2 +0 (Ω)]∗ and g ∈ H +1 +2 (∂Ω) be given. Then there exists a unique u ∈ Wδ,2(Ω) satisfying the +inhomogeneous Dirichlet boundary conditions (1.7) in the trace sense and the weak form (5.1). +Moreover, +(5.6) +∥u∥Wδ,2(Ω) ≤ C(d, ρ, Ω) +� +∥f∥[Wδ,2 +0 (Ω)]∗ + ∥g∥H +1 +2 (∂Ω) +� +. +Proof. Consider the equivalent weak form of the problem (5.1) with inhomogeneous Dirichlet +data: +Bρ,δ(w, v) = ⟨f − LδG, v⟩ , +∀v ∈ Wδ,2 +0 (Ω) . +By Theorem 3.9, LδG ∈ [Wδ,2 +0 (Ω)]∗, and so existence and uniqueness of w ∈ Wδ,2 +0 (Ω) follows +from Theorem 5.2. + +28 +JAMES M. SCOTT AND QIANG DU +Since u = w + G ∈ Wδ,2(Ω), we have by Theorem 3.9 +Bρ,δ(u, v) = Bρ,δ(w, v) + Bρ,δ(G, v) = ⟨f, v⟩ − ⟨LδG, v⟩ + Bρ,δ(G, v) += ⟨f, v⟩ − Bρ,δ(G, v) + Bρ,δ(G, v) = ⟨f, v⟩ +for any v ∈ Wδ,2 +0 (Ω), which is (5.1). +Moreover, by the assumption on G and Theorem 1.5 we have +∥G∥Wδ,2(Ω) ≤ C ∥G∥H1(Ω) ≤ C ∥g∥H +1 +2 (∂Ω) . +(5.7) +By (5.3), (3.20), and (5.7) +∥w∥Wδ,2(Ω) ≤ C +� +∥f∥[Wδ,2 +0 (Ω)]∗ + ∥LδG∥[Wδ,2 +0 (Ω)]∗ +� +≤ C +� +∥f∥[Wδ,2 +0 (Ω)]∗ + ∥G∥Wδ,2(Ω) +� +≤ C +� +∥f∥[Wδ,2 +0 (Ω)]∗ + ∥g∥H +1 +2 (∂Ω) +� +. +These estimates on G and w lead to (5.6). The equivalence of traces u = g on ∂Ω is established +by (5.6) and Theorem 4.2. Obviously, the solution u defined via (5.4) is independent of the +extension G chosen. +□ +5.3. The Neumann problem. The nonlocal Poisson problem with inhomogeneous Neumann +boundary conditions is given by (1.6)-(1.8). The special case of homogeneous boundary con- +ditions g = 0 is covered here as well. In step with the treatment for the classical Neumann +problem, we see that solutions of (1.6)-(1.8) are unique up to constants, and an application of +the nonlocal Green’s identity (1.5) shows that the compatibility condition +ˆ +Ω +f(x) dx + +ˆ +∂Ω +g(x) dσ(x) = 0 +is required for existence of a solution. +We introduce a weak formulation of (1.6)-(1.8) by the following formal computation: Sup- +pose that u ∈ C2(Ω), ∂u +∂ν = g on ∂Ω, and u satisfies Lδu = f for some given function f. Then +for arbitrary v ∈ C2(Ω), the nonlocal Green’s identity (1.5) gives +Bρ,δ(u, v) = ⟨Lδu, v⟩ + +ˆ +∂Ω +∂u +∂ν (x)v(x) dσ(x) = ⟨f, v⟩ + +ˆ +∂Ω +g(x)v(x) dσ(x) . +Recall the definition of Φδ = Φδ,0 as in (1.13) for α = 0. For a kernel ρ that satisfies (A1), +define the function space +˚ +Wδ,2 +ρ (Ω) := +� +u ∈ Wδ,2(Ω) : = ⟨Φδ, u⟩ = 0 = (Φδu)Ω +� +. +It is clear that ˚ +Wδ,2 +ρ (Ω) is a closed subspace of Wδ,2(Ω) with its inner product inherited from +Wδ,2(Ω). +Definition 5.4. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). For f ∈ [Wδ,2(Ω)]∗ and g ∈ +H− 1 +2(∂Ω) satisfying +(5.8) +⟨f, 1⟩ + ⟨g, 1⟩ = 0 , +we say that u ∈ ˚ +Wδ,2 +ρ (Ω) is a weak solution to (1.6)-(1.8) if +(5.9) +Bρ,δ(u, v) = ⟨f, v⟩ + ⟨g, Tv⟩ , +∀v ∈ ˚ +Wδ,2 +ρ (Ω) . +Theorem 5.5 (Well-posedness). Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). Let δ < δ0. Then +there exists a constant C = C(d, ρ, Ω, CN) such that +(5.10) +∥u∥2 +Wδ,2(Ω) ≤ CBρ,δ(u, u) , +∀u ∈ ˚ +Wδ,2 +ρ (Ω) . + +NONLOCAL BOUNDARY-VALUE PROBLEMS +29 +Moreover, for any f ∈ [Wδ,2(Ω)]∗ and g ∈ H− 1 +2(∂Ω) with (5.8) satisfied, there exists a unique +solution u ∈ ˚ +Wδ,2 +ρ (Ω) to (5.9) satisfying the energy estimate +(5.11) +∥u∥Wδ,2(Ω) ≤ ∥f∥[Wδ,2(Ω)]∗ + ∥g∥H− 1 +2 (∂Ω) . +Proof. The right-hand side of (5.9) defines an element of [Wδ,2(Ω)]∗ acting on v thanks to The- +orem 4.2. The coercivity (5.10) follows from (1.16) and Theorem 4.8. We conclude the existence +and uniqueness of a weak solution via the Lax-Milgram theorem thanks to the continuity of Bρ,δ +established in (1.17). +□ +6. Boundary-localized convolutions associated to the nonlocal operator +For α ≥ 0 and for a measurable function u : Ω → R, we recall the definition of the operator +Kδ,α and the function Φδ,α in (1.13). +We use the convention Kδ = Kδ,0. +(recall the same +convention Φδ = Φδ,0 was used in Section 5.3). +We define the adjoint operator K∗ +δ,α of Kδ,α by +(6.1) +K∗ +δ,αu(x) := +ˆ +Ω +1 +Φδ,α(y)ρδ,α(x, y)u(y) dy , +with the convention that K∗ +δ = K∗ +δ,0. Then as a distribution +⟨Kδ,αu, ϕ⟩ = +� +u, K∗ +δ,αϕ +� +and +� +K∗ +δ,αu, ϕ +� += ⟨u, Kδ,αϕ⟩ . +Theorem 6.1. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). Let α ∈ R, and let u ∈ L2(Ω). +Then +(6.2) +∥u − Kδ,αu∥2 +L2(Ω) ≤ +ˆ +Ω +ˆ +Ω +1 +Φδ,α(x)ρδ,α(x, y)|u(x) − u(y)|2 dy dx . +Consequently, there exists a constant C depending only on d, ρ, Ω and α such that +(6.3) +∥u − Kδ,αu∥L2(Ω) ≤ C min{δ, diam(Ω)2}[u]Wδ,2(Ω) , +∀u ∈ Wδ,2(Ω). +Proof. By definition of Φδ,α, Hölder’s inequality gives +∥u − Kδ,αu∥2 +L2(Ω) = +ˆ +Ω +1 +Φδ,α(x)2 +�ˆ +Ω +ρδ,α(x, y)(u(y) − u(x)) dy +�2 +dx +≤ +ˆ +Ω +1 +Φδ,α(x)2 +�ˆ +Ω +ρδ,α(x, y) dy +� �ˆ +Ω +ρδ,α(x, y)|u(y) − u(x)|2 dy +� +dx += +ˆ +Ω +ˆ +Ω +1 +Φδ,α(x)ρδ,α(x, y)|u(x) − u(y)|2 dy dx , +which is (6.2). Now, from (2.16) and Lemma 2.1 we have the estimate +ρδ,α(x, y) +Φδ,α(x) +≤ C(d, Ω, ρ, α)ρδ(x, y) ≤ C(d, Ω, ρ, α)ηδ(x)2ρδ,2(x, y) +in the right-hand side integral of (6.2), from which (6.3) follows. +□ +Theorem 6.2. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). Let α ≥ 2 and let u ∈ L2(Ω). Then +there exists C = C(d, ρ, Ω, α) > 0 such that +∥∇Kδ,αu∥L2(Ω) ≤ C +ˆ +Ω +ˆ +Ω +ρδ,2(x, y)|u(x) − u(y)|2 dy dx ++ C +ˆ +Ω +ˆ +Ω +ρδ,2(x, y)|u(x) − u(y)|2 dy dx . +(6.4) + +30 +JAMES M. SCOTT AND QIANG DU +Proof. Assume the right-hand side of (6.4) is finite. We have +∇Kδ,αu(x) = +1 +Φδ,α(x) +ˆ +Ω +∇xρδ,α(x, y)u(y) dy − ∇Φδ,α(x) +Φδ,α(x)2 +ˆ +Ω +ρδ,α(x, y)u(y) dy += +1 +Φδ,α(x) +ˆ +Ω +∇xρδ,α(x, y)(u(y) − u(x)) dy − ∇Φδ,α(x) +Φδ,α(x)2 +ˆ +Ω +ρδ,α(x, y)(u(y) − u(x)) dy , +where in the last line we added and subtracted ∇Φδ,α(x) +Φδ,α(x) u(x). Therefore, by Hölder’s inequality +∥∇Kδ,αu∥2 +L2(Ω) ≤ +ˆ +Ω +� ηδ(x) +Φδ,α(x) +ˆ +Ω +|∇xρδ,α(x, z)| dz +� ˆ +Ω +|∇xρδ,α(x, y)| +ηδ(x)Φδ,α(x) |u(y) − u(x)|2 dy dx ++ +ˆ +Ω +|∇Φδ,α(x)|2 +|Φδ,α(x)|2 +�ˆ +Ω +ρδ,α(x, z) +Φδ,α(x) dz +� ˆ +Ω +ρδ,α(x, y) +Φδ,α(x) |u(y) − u(x)|2 dy dx . +We use (2.18) and (2.16) in the first integral and the definition of Φδ,α and (2.19) in the second +integral to get +∥∇Kδ,αu∥2 +L2(Ω) ≤ C +ˆ +Ω +ˆ +Ω +|∇xρδ,α(x, y)| +ηδ(x)Φδ,α(x) |u(y) − u(x)|2 dy dx ++ C +ˆ +Ω +ˆ +Ω +ρδ,α(x, y) +ηδ(x)2Φδ,α(x)|u(y) − u(x)|2 dy dx . +(6.5) +From Theorem 2.6 we have +|∇xρδ,α(x, y)| ≤ ρδ,α+1(x, y) + C(d, α)ρδ(x),α+1(|y − x|) , +and so with (2.16) and Lemma 2.1 we have the estimates +|∇xρδ,α(x, y)| +ηδ(x)Φδ,α(x) ≤ Cρδ,2(x, y) + Cρδ,2(x, y) , +ρδ,α(x, y) +ηδ(x)2Φδ,α(x) ≤ Cρδ,2(x, y) . +Using the previous two estimates in (6.5) gives (6.4). +□ +As a consequence of the embedding and characterization properties of the nonlocal function +space proved in Theorem 1.5 and Theorem 1.7, we have the following corollary: +Corollary 6.3. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). Let α ≥ 2 and let u ∈ Wδ,2(Ω). +Then there exists C = C(d, ρ, Ω, α) > 0 such that +∥∇Kδ,αu∥L2(Ω) ≤ C[u]Wδ,2(Ω) . +(6.6) +Theorem 6.4. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). Let α ≥ 0. There exists a constant +C = C(d, ρ, Ω, α) > 0 such that +(6.7) +∥ηδKδ,αu∥L2(Ω) + +��η2 +δ∇Kδ,αu +�� +L2(Ω) ≤ C ∥u∥H−1(Ω) +∀u ∈ H−1(Ω) , +(6.8) +∥Kδ,αu∥L2(Ω) + ∥ηδ∇Kδ,αu∥L2(Ω) ≤ C ∥u∥L2(Ω) +∀u ∈ L2(Ω) , +and +(6.9) +∥∇Kδ,αu∥L2(Ω) ≤ C ∥∇u∥L2(Ω) +∀u ∈ H1(Ω) , +Proof. We first prove (6.8). By Hölder’s inequality, Tonelli’s theorem, (2.16) and Corollary 2.4 +∥Kδ,αu∥2 +L2(Ω) ≤ +ˆ +Ω +�´ +Ω ρδ,α(x, z) dz +Φδ,α(x) +� � +1 +Φδ,α(x) +ˆ +Ω +ρδ,α(x, y)|u(y)|2 dy +� +dx ≤ C ∥u∥2 +L2(Ω) . +(6.10) + +NONLOCAL BOUNDARY-VALUE PROBLEMS +31 +We estimate ηδ∇Kδ,αu similarly, additionally using (2.19) and (2.18): +∥ηδ∇Kδ,αu∥2 +L2(Ω) +≤ +ˆ +Ω +� +ηδ(x) +´ +Ω |∇xρδ,α(x, y)| dy +Φδ,α(x) +� � ηδ(x) +Φδ,α(x) +ˆ +Ω +|∇xρδ,α(x, y)| |u(y)|2 dy +� +dx ++ +ˆ +Ω +�´ +Ω ρδ,α(x, y) dy +Φδ,α(x) +� � +1 +Φδ,α(x) +ˆ +Ω +ρδ,α(x, y)|u(y)|2 dy +� +dx ≤ C ∥u∥2 +L2(Ω) . +(6.11) +Then (6.10)-(6.11) establish (6.8). +Next we prove (6.7). We first recall the representation of distributions in H−1(Ω). For any +u ∈ H−1(Ω) there exist u0 ∈ L2(Ω) and u1 ∈ L2(Ω; Rd) such that +(6.12) +⟨u, ϕ⟩ = ⟨u0, ϕ⟩ + ⟨u1, ∇ϕ⟩ , +∀ϕ ∈ H1 +0(Ω) , +with +(6.13) +∥u∥2 +H−1(Ω) = inf +� +∥u0∥2 +L2(Ω) + ∥u1∥2 +L2(Ω) : u0 and u1 satisfy (6.12) +� +. +Now let ϕ ∈ L2(Ω) be arbitrary, and define +ϕδ,α(x) := +ˆ +Ω +ρδ,α(x, y) ηδ(y) +Φδ,α(y)ϕ(y) dy . +Then ϕδ,α ∈ C1(Ω), and estimates similar to (6.10)-(6.11) reveal that ϕδ,α ∈ H1(Ω) with +(6.14) +∥ϕδ,α∥H1(Ω) ≤ C ∥ϕ∥L2(Ω) . +Furthermore, if ϕ ∈ C1(Ω) then by (2.2)-(2.3) and Lemma 2.3 +|ϕδ,α(x)| ≤ C(ρ, Ω)ηδ(x) +ˆ +Ω +ρηδ(y)(|y − x|)ϕ(y) dy ≤ C ∥ϕ∥L∞(Ω) ηδ(x) , +and so +Tϕδ,α ≡ 0 , +∀ϕ ∈ C1(Ω) . +Let ϕn be a sequence in C1(Ω) converging to ϕ in L2(Ω). Then by (6.14) +∥Tϕδ,α∥H1/2(∂Ω) ≤ ∥Tϕδ,α − T(ϕn)δ,α∥H1/2(∂Ω) +≤ C ∥ϕδ,α − (ϕn)δ,α∥H1(Ω) +≤ C ∥ϕ − ϕn∥L2(Ω) → 0 as n → ∞ . +Therefore +ϕδ,α ∈ H1 +0(Ω) , +∀ϕ ∈ L2(Ω) . +Now let u ∈ H−1(Ω). For arbitrary ϕ ∈ L2(Ω) we will use (6.12) with test function ϕδ,α to +obtain that ηδKδ,αu actually defines a function in L2(Ω). To begin, let u0 and u1 satisfy (6.12), +and write +(6.15) +⟨ηδKδ,αu, ϕ⟩ = ⟨u, ϕδ,α⟩ = +ˆ +Ω +u0ϕδ,α dx + +ˆ +Ω +u1 · ∇(ϕδ,α) dx . + +32 +JAMES M. SCOTT AND QIANG DU +Using (2.2)-(2.3), (2.17), and additionally Hölder’s inequality and Lemma 2.3, +ˆ +Ω +ˆ +Ω +|u0(x)| ηδ(y) +Φδ,α(y)ρδ,α(x, y)|ϕ(y)| dx dy ++ +ˆ +Ω +ˆ +Ω +|u1(x)| ηδ(y) +Φδ,α(y)|∇xρδ,α(x, y)||ϕ(y)| dx dy +≤ C +ˆ +Ω +ˆ +Ω +� +ρδ,0(x, y) + ρδ,0(x, y) +�� +|u0(x)| + |u1(x)| +� +|ϕ(y)| dy dx +≤ C +�ˆ +Ω +ˆ +Ω +� +ρδ,0(x, y) + ρδ,0(x, y) +� +|u0(x)|2 dy dx +�1/2 +�ˆ +Ω +ˆ +Ω +� +ρδ,0(x, y) + ρδ,0(x, y) +� +|ϕ(y)|2 dy dx +�1/2 ++ C +�ˆ +Ω +ˆ +Ω +� +ρδ,0(x, y) + ρδ,0(x, y) +� +|u1(x)|2 dy dx +�1/2 +�ˆ +Ω +ˆ +Ω +� +ρδ,0(x, y) + ρδ,0(x, y) +� +|ϕ(y)|2 dy dx +�1/2 +≤ C +� +∥u0∥L2(Ω) + ∥u1∥L2(Ω) +� +∥ϕ∥L2(Ω) . +(6.16) +So can apply Fubini’s theorem in (6.15) to obtain the identity +⟨ηδKδ,αu, ϕ⟩ = +ˆ +Ω +� ηδ(x) +Φδ,α(x) +ˆ +Ω +ρδ,α(x, y)u0(y) dy +� +ϕ(x) dx ++ +ˆ +Ω +� ηδ(x) +Φδ,α(x) +ˆ +Ω +∇yρδ,α(x, y)u1(y) dy +� +ϕ(x) dx . +The estimate (6.16) shows that this identity is independent of the choice of u0 and u1, and so +ηδKδ,αu defines a measurable function on Ω. Further, by (6.16) and (6.13) +∥ηδKδ,αu∥L2(Ω) = +sup +∥ϕ∥L2(Ω)≤1 +⟨ηδKδ,αu, ϕ⟩ ≤ C ∥u∥H−1(Ω) , +which is the first half of (6.7). For the other half, since for any ϕ ∈ C∞ +c (Ω) +� +η2 +δ∇Kδ,αu, ϕ +� += ⟨u, ¯ϕδ,α⟩ , +where ¯ϕδ,α(x) = +ˆ +Ω +∇y +�ρδ,α(x, y) +Φδ,α(y) +� +η2 +δ(y)ϕ(y) dy , +the estimate +��η2 +δ∇Kδ,αu +�� +L2(Ω) ≤ C ∥u∥H−1(Ω) is established using a similar process. +Finally, by Theorem 6.2, in order to establish (6.9) it suffices to show +ˆ +Ω +ˆ +Ω +ρδ,2(x, y)|u(x) − u(y)|2 dy dx ++ +ˆ +Ω +ˆ +Ω +ρδ,2(x, y)|u(x) − u(y)|2 dy dx ≤ C ∥∇u∥2 +L2(Ω) . +But this is a consequence of the embedding and characterization properties proved in Theo- +rem 1.5 and Theorem 1.7, additionally using the upper bound on ρ. +□ +Lemma 6.5. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). Suppose that ϕ ∈ L2(Ω) has compact +support in Ω, i.e. there exists r(ϕ) > 0 such that +supp ϕ ⊂ {x ∈ Ω : dist(x, ∂Ω) > +� +r(ϕ)} = Rd \ Ω√ +r(ϕ) . +Then Kδ,αϕ has compact support with +(6.17) +supp Kδ,αϕ ⊂ +� +x ∈ Ω : dist(x, ∂Ω) > 1 − ¯κ0R0 +√ +δ +1 + ¯κ0R0 +√ +δ +� +r(ϕ) +� +. + +NONLOCAL BOUNDARY-VALUE PROBLEMS +33 +Proof. By Lemma 2.2 whenever dist(x, ∂Ω) < 1−¯κ0R0 +√ +δ +1+¯κ0R0 +√ +δ +� +r(ϕ) we have +{y : |x − y| ≤ R0ηδ(x)} ⊂ Ω√ +r(ϕ) and {y : |x − y| ≤ R0ηδ(y)} ⊂ Ω√ +r(ϕ) . +Therefore, the domains of integration in the integrals defining Kδ,αϕ and supp ϕ are disjoint, +and Kδ,αϕ(x) = 0. +□ +Theorem 6.6. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). Let α ≥ 0. There exists a constant +C = C(d, ρ, Ω, α) > 0 such that +(6.18) +��K∗ +δ,αu +�� +H−1(Ω) + +��ηδK∗ +δ,αu +�� +L2(Ω) + +��η2 +δ∇K∗ +δ,αu +�� +L2(Ω) ≤ C ∥u∥H−1(Ω) , +∀u ∈ H−1(Ω) +and +(6.19) +��K∗ +δ,αu +�� +L2(Ω) + +��ηδ∇K∗ +δ,αu +�� +L2(Ω) ≤ C ∥u∥L2(Ω) , +∀u ∈ L2(Ω) . +Proof. We only prove +(6.20) +��K∗ +δ,αu +�� +H−1(Ω) ≤ C ∥u∥H−1(Ω) ; +the estimates for ηδK∗ +δ,αu and η2 +δ∇K∗ +δ,αu are established similarly to (6.7). +It is a consequence of Lemma 6.5 that +T[Kδ,αϕ] ≡ 0 , +∀ϕ ∈ C∞ +c (Ω) . +Thanks to the continuity of the operator Kδ,α with respect to the H1(Ω)-convergence implied by +(6.8)-(6.9), and thanks to continuity in H1(Ω) of the trace operator, we conclude that Kδ,αϕ ∈ +H1 +0(Ω) whenever ϕ ∈ H1 +0(Ω). Hence +(6.21) +| +� +ϕ, K∗ +δ,αu +� +H1 +0(Ω),H−1(Ω) | = | ⟨u, Kδ,αϕ⟩H1 +0(Ω),H−1(Ω) | ≤ C ∥u∥H−1(Ω) ∥Kδ,αϕ∥H1(Ω) , +and so by (6.8) and (6.9) we conclude (6.20). +The proof of (6.19) is similar to the proof of (6.8). +□ +6.1. Convergence in the localization limit. +Theorem 6.7. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). Let α ≥ 0 and u ∈ L2(Ω). Then +(6.22) +lim +δ→0 ∥Kδ,αu − u∥L2(Ω) = 0 +and +(6.23) +lim +δ→0 +��K∗ +δ,αu − u +�� +L2(Ω) = 0 . +Proof. First we prove (6.22). It suffices to show that +(6.24) +Kδ,αu(x) → u(x) almost everywhere in Ω , +since then (6.8) with the dominated convergence theorem implies (6.22). +Fix a Lebesgue point x ∈ Ω of u, so that +lim +r→0 + +B(x,r) +|u(y) − u(x)| dy = 0 +holds. Choose ¯δ0 depending on x such that +(6.25) +dist(x, ∂Ω) ≥ +1 +√κ0 +1 + ¯κ0R0 +�¯δ0 +1 − ¯κ0R0 +�¯δ0 +�¯δ0 . +Then (2.6) and (2.7) hold and x ∈ Rd \ Ω√ +δ/κ0 for all δ < ¯δ0. Therefore by definition of ηδ +Φδ,α(x) = +ˆ +B(x,R0δ) +1 +δd+α ρ +�|y − x| +δ +� +dy = ¯ρδ−α , +∀δ < ¯δ0 + +34 +JAMES M. SCOTT AND QIANG DU +and so +Kδ,αu(x) = +1 +2Φδ,α(x) +� ˆ +{|y−x|≤R0ηδ(x)}∩(Rd\Ω√ +δ/κ0) +1 +ηδ(x)d+α ρ +�|y − x| +ηδ(x) +� +u(y) dy ++ +ˆ +{|y−x|≤R0ηδ(x)}∩(Rd\Ω√ +δ/κ0) +1 +ηδ(y)d+α ρ +�|y − x| +ηδ(y) +� +u(y) dy +� += δα +¯ρ +ˆ +B(x,R0δ) +1 +δd+α ρ +�|y − x| +δ +� +u(y) dy += +ˆ +B(x,δ) +1 +δd ρ1 +�y − x +δ +� +u(y) dy , +∀δ < ¯δ0 , +where, for ¯ρ given by (6.28) below, ρ1(x) := ρ(|x|) +¯ρ +is a continuous function with support in +B(0, 1) and satisfies ∥ρ1∥L1(Rd) = 1. So for δ < ¯δ0 we have +|Kδ,αu(x) − u(x)| ≤ +ˆ +B(x,δ) +1 +δd ρ1 +�y − x +δ +� +|u(y) − u(x)| dy ≤ C + +B(x,δ) +|u(y) − u(x)| dy → 0 +as δ → 0. This limit holds for all Lebesgue points x ∈ Ω of u, that is, almost everywhere in Ω, +so (6.24) is established. +The proof of (6.23) follows the same argument, with (6.19) in place of (6.8). +□ +It is straightforward to prove the following corollary using the same method: +Corollary 6.8. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1). Let α ≥ 0 and u ∈ H1(Ω). Then +(6.26) +lim +δ→0 ∥Kδ,αu − u∥H1(Ω) = 0 +and +(6.27) +lim +δ→0 +��K∗ +δ,αu − u +�� +H1(Ω) = 0 . +6.2. Application: proofs of the Poincaré inequalities. +Lemma 6.9. Let Ω satisfy (AΩ) and ρ satisfy (A1). Define the positive constant ¯ρ by +(6.28) +¯ρ := +ˆ +B(0,1) +ρ(|z|) dz . +Then +lim +δ→0 ∥Φδ − ¯ρ∥Lp(Ω) = 0 , +for any 1 ≤ p < ∞ . +Proof. By (2.16) and the dominated convergence theorem it suffices to show that +Φδ(x) → ¯ρ almost everywhere in Ω . +Fix x ∈ Ω. Choose ¯δ0 depending on x such that (6.25) holds. Then (2.6) and (2.7) hold and +x ∈ Rd \ Ω√ +δ/κ0 for all δ < ¯δ0. Therefore by definition of ηδ +Φδ(x) = +ˆ +B(x,R0δ) +1 +δd ρ +�|y − x| +δ +� +dy = ∥ρ∥L1(B(0,R0)) , +∀δ < ¯δ0 . +The conclusion follows. +□ +proof of Theorem 4.7. Note that Kδu ∈ H1(Ω) by Corollary 6.3 and (6.8). +Let {un} be a +sequence in C∞ +c (Ω) converging to u in Wδ,2(Ω). Then by the classical trace theorem for Sobolev +functions, and again by Corollary 6.3 and (6.8) +∥T(Kδun) − T(Kδu)∥H +1 +2 (∂Ω) ≤ C ∥Kδun − Kδu∥H1(Ω) ≤ C ∥un − u∥Wδ,2(Ω) . + +NONLOCAL BOUNDARY-VALUE PROBLEMS +35 +However, by Lemma 6.5 +T(Kδun) = 0 , +∀n . +It follows that the H +1 +2 (∂Ω)-norm of T(Kδu) can be made arbitrarily small by choosing n suffi- +ciently large; therefore +T(Kδu) = 0 on ∂Ω . +Hence Kδu ∈ H1 +0(Ω), and we can apply the classical Poincaré inequality: +(6.29) +∥Kδu∥L2(Ω) ≤ C(Ω) ∥∇Kδu∥L2(Ω) . +By Corollary 6.3 +∥Kδu∥L2(Ω) ≤ C(d, ρ, Ω)[u]Wδ,2(Ω) . +Finally by (6.3) +∥u∥L2(Ω) ≤ ∥Kδu∥L2(Ω) + ∥u − Kδu∥L2(Ω) +≤ (C + C min{δ, diam(Ω)2})[u]Wδ,2(Ω) . +□ +proof of Theorem 4.8. First, by the local Poincaré inequality, there exists a constant ¯C = +¯C(d, ρ, Ω, δ) > 0 such that +(6.30) +∥v∥L2(Ω) ≤ ¯C ∥∇v∥L2(Ω) +for all v ∈ H1(Ω) with (Φδv)Ω = 0. +Next, we show that in fact the constant ¯C can be made independent of δ. +Define the +Rayleigh quotient +λδ := min +�´ +Ω |∇u(x)|2 dx +´ +Ω |u(x)|2 dx +: u ∈ H1(Ω) with (Φδu)Ω = 0 +� +. +By (6.30), λδ > 0 for all δ > 0. We claim that +(6.31) +inf +δ>0 λδ > 0 . +Suppose to the contrary; then there exists a subsequence (not relabeled) δ → 0 and a sequence +of normalized eigenfunctions {uδ}δ for which the minimums λδ are attained with +∥uδ∥L2(Ω) = 1 , +(Φδuδ) = 0 , +∥∇uδ∥L2(Ω) → 0 as δ → 0 . +Then there exists a function u ∈ H1(Ω) such that uδ converges strongly to u in L2(Ω) and +weakly to u in H1(Ω). Moreover, u(x) = c, a constant, in Ω. However, by Lemma 6.9 +c¯ρ = + +Ω +u(x)¯ρ dx = lim +δ→0 + +Ω +uδ(x)Φδ(x) dx = 0 . +Therefore it must be that c = 0, a contradiction since ∥uδ∥L2(Ω) = 1. We then conclude from +(6.31) that for any δ < δ0 +(6.32) +∥v∥L2(Ω) ≤ C(d, ρ, Ω) ∥∇v∥L2(Ω) , +∀v ∈ H1(Ω) with (Φδv)Ω = 0 . +Finally, let u ∈ Wδ,2(Ω) with (Φδu)Ω = 0. Corollary 6.3 and (6.8) imply that Kδu ∈ H1(Ω), +and we additionally have +ˆ +Ω +Φδ(x)Kδu(x) dx = +ˆ +Ω +ˆ +Ω +ρδ(x, y)u(y) dy dx = +ˆ +Ω +Φδ(y)u(y) dy = 0 . +Applying the local, and uniform in δ, Poincaré inequality established in (6.32) to Kδu, we get: +∥Kδu∥L2(Ω) ≤ C(d, ρ, Ω) ∥∇Kδu∥L2(Ω) . +From here the proof proceeds identically to the proof of Theorem 4.7, (6.29). +□ + +36 +JAMES M. SCOTT AND QIANG DU +6.3. Application: +regularity results for nonlocal problems. As an application of the +estimates for the boundary-localized convolutions, we obtain some regularity results for the +weak solutions of nonlocal problems. Let us examine the case of the Dirichlet problem: +Theorem 6.10. Let Ω ⊂ Rd satisfy (AΩ). Suppose that u ∈ Wδ,2(Ω) is a weak solution of (1.6) +with Dirichlet data, i.e. u satisfies (1.7) and (5.1), where additionally f ∈ H1 +loc(Ω). Then there +exists C depending only on d, Ω and ρ such that +(6.33) +∥u∥H1(Ω) ≤ C +� +∥f∥[Wδ,2 +0 (Ω)]∗ + ∥g∥H +1 +2 (∂Ω) + ∥ηδf∥L2(Ω) + +��η2 +δ∇f +�� +L2(Ω) +� +. +In particular, u ∈ H1(Ω) whenever the right-hand side of (6.33) is finite. +Proof. Let ϕ ∈ C∞ +c (Ω). By Proposition 4.5, case i), we conclude that the nonlocal Green’s +identity (1.5) holds for u and ϕ; that is, +(6.34) +Bρ,δ(u, ϕ) = +ˆ +Ω +Lδu(x)ϕ(x) dx , +since the boundary term is 0. Therefore, if f ∈ H1 +loc(Ω) then by (5.1) +⟨f, ϕ⟩ = +ˆ +Ω +f(x)ϕ(x) dx = +ˆ +Ω +Lδu(x)ϕ(x) dx , +∀ϕ ∈ C∞ +c (Ω) . +Both Lδu and f are locally integrable, so the nonlocal equation (1.6) holds pointwise for almost +every x ∈ Ω, and it follows that +(6.35) +u(x) = Kδ,2u(x) + +1 +2 · Φδ,2(x)f(x) , +for almost every x ∈ Ω . +Then by Corollary 6.3 and (5.6) +∥∇Kδ,2u∥2 +L2(Ω) ≤ C[u]2 +Wδ,2(Ω) ≤ C +� +∥f∥2 +[Wδ,2 +0 (Ω)]∗ + ∥g∥2 +H +1 +2 (∂Ω) +� +. +Next, we know that +(6.36) +∇ +� +1 +2 · Φδ,2(x)f(x) +� += +1 +2 · Φδ,2(x)∇f(x) − ∇Φδ,2(x) +2 · Φδ,2(x)2 f(x) , +and so taking the L2(Ω)-norm on both sides and using the bounds on Φδ,2, its reciprocal and its +derivatives, +(6.37) +����∇ +� f +Φδ,2 +����� +L2(Ω) +≤ C +��η2 +δ∇f +�� +L2(Ω) + C ∥ηδf∥L2(Ω) . +Combining the previous three estimates with the pointwise equality (1.6) and using (5.6) to +estimate the L2(Ω)-norm of u, we conclude +∥u∥H1(Ω) ≤ C +� +∥f∥[Wδ,2 +0 (Ω)]∗ + ∥g∥H +1 +2 (∂Ω) + +��η2 +δ∇f +�� +L2(Ω) + ∥ηδf∥L2(Ω) +� +. +□ +We can prove a statement of local regularity in a similar way: +Theorem 6.11. Let Ω ⊂ Rd satisfy (AΩ), and let x0 ∈ Ω. Suppose that f ∈ [Wδ,2 +0 (Ω)]∗ has the +property that for every ε ∈ (0, dist(x0, ∂Ω)) it agrees with a function h ∈ H1(Ω \ Bε(x0)), i.e. +⟨f, v⟩ = +ˆ +Ω +h(x)v(x) dx , +∀v ∈ H1 +0(Ω) with suppv ⊂ Ω \ Bε(x0) . +Suppose u is a weak solution of the inhomogeneous Dirichlet problem, i.e. u satisfies (1.7) and +(5.1) with Poisson data f. Then u ∈ H1(Ω \ Bε(x0)) for every ε > 0. + +NONLOCAL BOUNDARY-VALUE PROBLEMS +37 +Proof. Let ϕ ∈ C∞ +c (Ω \ Bε(x0)) be arbitrary. Then by Proposition 4.5, case i), the nonlocal +Green’s identity (1.5) holds: +ˆ +Ω +h(x)ϕ(x) dx = Bρ,δ(u, ϕ) = +ˆ +Ω +Lδu(x)ϕ(x) dx . +Therefore +Lδu(x) = h(x) +for almost every x ∈ Ω \ Bε(x0) . +So we see that the formula +u(x) = Kδ,2u(x) + +1 +2 · Φδ,2(x)h(x) +holds for almost every x ∈ Ω \ Bε(x0). +By Corollary 6.3 and (5.6) +∥∇Kδ,2u∥2 +L2(Ω) ≤ C[u]2 +Wδ,2(Ω) ≤ C +� +∥f∥2 +[Wδ,2 +0 (Ω)]∗ + ∥g∥2 +H +1 +2 (∂Ω) +� +. +Next, we know that +∇ +� +1 +2 · Φδ,2(x)h(x) +� += +1 +2 · Φδ,2(x)∇h(x) − ∇Φδ,2(x) +2 · Φδ,2(x)2 h(x) , +and so taking the L2(Ω\Bε(x0))-norm on both sides and using the bounds on Φδ,2, its reciprocal +and its derivatives, +����∇ +� h +Φδ,2 +����� +L2(Ω\Bε(x0)) +≤ C +��η2 +δ∇h +�� +L2(Ω\Bε(x0)) + C ∥ηδh∥L2(Ω\Bε(x0)) . +Combining the previous three estimates with the pointwise equality, we conclude +∥∇u∥L2(Ω\Bε(x0)) ≤ C +� +∥f∥[Wδ,2 +0 (Ω)]∗ + +��η2 +δ∇h +�� +L2(Ω\Bε(x0)) + ∥ηδh∥L2(Ω\Bε(x0)) +� +. +□ +Next, we present a similar regularity result for Neumann problems. To do this, some changes +must be made in order to account for the boundary data. +Theorem 6.12. Let f ∈ [Wδ,2(Ω)]∗ and g ∈ H− 1 +2(∂Ω) with ⟨f, 1⟩ + ⟨g, 1⟩ = 0. Suppose that +u ∈ ˚ +Wδ,2 +ρ (Ω) is a weak solution of the inhomogeneous Neumann problem, i.e. u satisfies (5.9) +where additionally f ∈ H1 +loc(Ω). Then there exists C depending only on Ω and ρ such that +(6.38) +∥u∥H1(Ω) ≤ C +� +∥f∥[Wδ,2(Ω)]∗ + ∥g∥H− 1 +2 (∂Ω) + ∥ηδf∥L2(Ω) + +��η2 +δ∇f +�� +L2(Ω) +� +. +In particular, u ∈ H1(Ω) whenever the right-hand side of the above inequality is finite. +Proof. Let ϕ ∈ C∞ +c (Ω). By Proposition 4.5, case i), we conclude that the nonlocal Green’s +identity (1.5) holds for u and ϕ; that is, +(6.39) +Bρ,δ(u, ϕ) = +ˆ +Ω +Lδu(x)ϕ(x) dx , +since the boundary term is 0. +Now for ϕ ∈ C∞ +c (Ω) arbitrary, define ψ(x) = ϕ(x) − (Φδϕ)Ω. Then we can use ψ as a test +function in (5.9) to get +ˆ +Ω +Lδu(x)ϕ(x) dx = Bρ,δ(u, ϕ) = Bρ,δ(u, ψ) = ⟨f, ψ⟩ + ⟨g, ψ⟩ += ⟨f, ϕ⟩ + ⟨g, ϕ⟩ − (Φδϕ)Ω (⟨f, 1⟩ + ⟨g, 1⟩) = ⟨f, ϕ⟩ . + +38 +JAMES M. SCOTT AND QIANG DU +So if f ∈ H1 +loc(Ω) then +ˆ +Ω +Lδu(x)ϕ(x) dx = +ˆ +Ω +f(x)ϕ(x) dx +∀ϕ ∈ C∞ +c (Ω) . +Both Lδu and f are locally integrable, so (6.35) holds almost everywhere in Ω. Proceeding +exactly as in the proof of Theorem 6.10 but with (5.11) in place of (5.6) we come to +∥u∥H1(Ω) ≤ C +� +∥f∥[Wδ,2(Ω)]∗ + ∥g∥H− 1 +2 (∂Ω) + +��η2 +δ∇f +�� +L2(Ω) + ∥ηδf∥L2(Ω) +� +. +□ +7. Consistency with classical boundary-value problems +Our aim is to show the consistency of the nonlocal boundary value problems with their +local limits without assuming extra regularity on the data and solutions than those established +earlier. To this end, we first examine the local limits of the nonlocal operators and the nonlocal +bilinear forms in suitable function spaces. +7.1. The operator in the localization limit. +Theorem 7.1. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1)-(A2). Let u ∈ W 2,p(Ω) for some +p ∈ [1, ∞). Then +lim +δ→0 +ˆ +Ω +|Lδu(x) − (−∆u(x))|p dx = 0 . +Proof. By Theorem 3.3 it suffices to prove the theorem for u ∈ C2(Ω). +Fix x ∈ Ω. Choose ¯δ0 depending on x such that (6.25) holds. Then for all δ < ¯δ0, (2.6) and +(2.7) hold and x ∈ Rd \ Ω√ +δ/κ0. Therefore by definition of ηδ +Lδu(x) = +ˆ +B(x,R0δ) +2 +δd+2 ρ +�|x − y| +δ +� +(u(x) − u(y)) dy +∀δ < ¯δ0 . +Taylor expanding, we have +− Lδu(x) = +ˆ +B(x,R0δ) +2 +δd+2 ρ +�|x − y| +δ +� +(y − x) dy · ∇u(x) ++ +ˆ +B(x,R0δ) +2 +δd+2 ρ +�|x − y| +δ +� ˆ 1 +0 +� +∇2u(y + t(x − y))(x − y), (x − y) +� +(1 − t) dt dy += +ˆ +B(0,R0) +2ρ (|z|) +ˆ 1 +0 +� +∇2u(x + (1 − t)δz)z, z +� +(1 − t) dt dz . +Noting that +2 +ˆ 1 +0 +(1 − t) dt +ˆ +B(0,R0) +ρ(|z|) +� +∇2u(x)z, z +� +dz = ∆u(x) , +we get +|Lδu(x) − (−∆u(x))| += +����� +ˆ +B(0,R0) +ˆ 1 +0 +2(1 − t)ρ (|z|) +�� +∇2u(x + (1 − t)δz) − ∇2u(x) +� +z, z +� +dt dz +����� . +Since ∇2u is continuous on Ω, by the dominated convergence theorem, we get as δ → 0 that +|Lδu(x) − (−∆u(x))| → 0 for almost every x ∈ Ω. +By Lemma 3.2, |Lδu(x) − (−∆u(x))|p is majorized independently of δ by a function be- +longing to L1(Ω), and so we obtain the result by again applying the dominated convergence +theorem. +□ + +NONLOCAL BOUNDARY-VALUE PROBLEMS +39 +7.2. The bilinear form in the localization limit. +Theorem 7.2. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1)-(A2). Then for any u, v ∈ H1(Ω) +lim +δ→0 Bρ,δ(u, v) = +ˆ +Ω +∇u(x) · ∇v(x) dx . +Proof. The proof is similar to Theorem 7.1. By Theorem 1.5 it suffices to prove the theorem for +u, v ∈ C2(Ω). Fix x ∈ Ω. Choose ¯δ0 depending on x such that (6.25) holds. Then for all δ < ¯δ0, +(2.6) and (2.7) hold and x ∈ Rd \ Ω√ +δ/κ0. Therefore by definition of ηδ +ˆ +Ω +ρδ,2 (x, y) (u(x) − u(y))(v(x) − v(y)) dy += +ˆ +B(x,R0δ) +1 +δd+2 ρ +�|x − y| +δ +� +(u(x) − u(y))(v(x) − v(y)) dy +∀δ < ¯δ0 . +Taylor expanding and then using (A2) and (2.1), we have +ˆ +Ω +ρδ,2 (x, y) (u(x) − u(y))(v(x) − v(y)) dy += +��ˆ +B(x,R0δ) +1 +δd ρ +�|x − y| +δ +� (y − x) ⊗ (y − x) +δ2 +dy +� +∇u(x), ∇v(x) +� ++ O +�ˆ +B(x,R0δ) +1 +δd+2 ρ +�|x − y| +δ +� +|y − x|3 dy +� += ∇u(x) · ∇v(x) + O(δ) . +Since ∇2u is continuous on Ω, the result follows by the dominated convergence theorem after +integrating both sides in x. +□ +Theorem 7.3. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1)-(A2). Suppose that v ∈ H1(Ω), and +suppose that {uδ} ⊂ H1(Ω) and u ∈ H1(Ω) satisfy uδ ⇀ u in H1(Ω) as δ → 0. Then +lim +δ→0 Bρ,δ(uδ − u, v) = 0 . +Proof. Without loss of generality assume uδ ⇀ 0 in H1(Ω) as δ → 0. First take v ∈ C2(Ω). +Then by the nonlocal Green’s identity Proposition 4.5, case ii), +Bρ,δ(uδ, v) = +ˆ +Ω +Lδv(x)uδ(x) dx + +ˆ +∂Ω +uδ(x) ∂v +∂ν (x) dσ(x) . +By Theorem 3.3 and Theorem 7.1 we obtain Lδv → −∆v strongly in L2(Ω) as δ → 0, and by +the compact embedding of H1(Ω) into L2(Ω) we have uδ → 0 strongly in L2(Ω). So, the first +term on the right-hand side converges to 0 as δ → 0. The second term on the right-hand side +converges to 0 as well, by the weak continuity of traces from H1(Ω) to H +1 +2(∂Ω). Therefore +lim +δ→0 Bρ,δ(uδ, v) = 0 +∀v ∈ C2(Ω) . +Now let v ∈ H1(Ω). Let {vn} be a sequence in C2(Ω) that converges to v in H1(Ω). Then +by Hölder’s inequality and Theorem 1.5 +|Bρ,δ(uδ, v)| ≤ |Bρ,δ(uδ, vn − v)| + |Bρ,δ(uδ, vn)| +≤ C ∥vn − v∥H1(Ω) ∥uδ∥H1(Ω) + |Bρ,δ(uδ, vn)| . +Since ∥uδ∥H1(Ω) is bounded uniformly with respect to δ, we can choose n large enough so that +the first term is arbitrarily small independent of δ. Then we use the first part of the proof to let +δ → 0 in the second term and obtain that +lim sup +δ→0 +|Bρ,δ(uδ, v)| = 0 +∀v ∈ H1(Ω) . + +40 +JAMES M. SCOTT AND QIANG DU +□ +7.3. The Dirichlet problem. For this section we consider δ varying, and analyze the case +δ → 0. Note that while an element of H−1(Ω) can serve as the Poisson data f for the local +problem (1.15), a properly regularized version, denoted by fδ, has to be used for a well-posed +nonlocal problem. We first show that the sequence of solutions to the nonlocal problem with the +regularized fδ converges as δ → 0 to the unique variational solution of (1.15) with the Poisson +data f, all subject to the same inhomogeneous Dirichlet boundary conditions (1.7). We then +illustrate how the regularization can be obtained. Further, we present the consistency of the +outward normal derivative on ∂Ω. +Theorem 7.4. Let Ω ⊂ Rd satisfy (AΩ). Let f ∈ H−1(Ω) and let g ∈ H +1 +2 (∂Ω). Suppose for +each δ there exists a distribution fδ that satisfies +(7.1) +lim +δ→0+ ⟨fδ − f, v⟩H−1(Ω),H1 +0(Ω) = 0 +∀v ∈ H1 +0(Ω) , +and that there exists C0 > 0 independent of δ such that +(7.2) +� +∥fδ∥[Wδ,2 +0 (Ω)]∗ + ∥ηδfδ∥L2(Ω) + +��η2 +δ∇fδ +�� +L2(Ω) +� +≤ C0 ∥f∥H−1(Ω) . +Then the corresponding solutions uδ of +(7.3) +Bρ,δ(uδ, v) = ⟨fδ, v⟩ , +∀v ∈ Wδ,2 +0 (Ω) , +with Poisson data fδ and Dirichlet boundary data g satisfy +(7.4) +∥uδ∥H1(Ω) ≤ C(d, Ω, ρ) +� +C0 ∥f∥H−1(Ω) + ∥g∥H +1 +2 (∂Ω) +� +. +Proof. The result follows from (6.33) and (7.2). +□ +Theorem 7.5. Let Ω ⊂ Rd satisfy (AΩ). Suppose that f ∈ H−1(Ω), and let fδ be a sequence +of functions that satisfy (7.1)-(7.2). Then the solutions uδ ∈ H1 +0(Ω) to (7.3) converge weakly in +H1(Ω) to a function u ∈ H1 +0(Ω), where u is the unique weak solution to the Poisson equation +(1.15) with Dirichlet boundary conditions (1.7). +Proof. Since the solutions satisfy the uniform H1 bound (7.4), it follows that they converge +weakly in H1(Ω) to a function u. By weak continuity of traces in H1(Ω), since Tuδ ≡ g for all +δ we have that Tu = g as well. +We now prove that u solves the weak form of the Poisson equation, i.e. +(7.5) +ˆ +Ω +∇u · ∇v dx = ⟨f, v⟩ , +∀v ∈ H1 +0(Ω) . +For each δ > 0, +⟨fδ, v⟩ = Bρ,δ(uδ, v) . +On one hand, (7.1) implies +lim +δ→0 ⟨fδ, v⟩ = ⟨f, v⟩ , +and on the other hand, Theorem 7.3 and Theorem 7.2 imply that +lim +δ→0 Bρ,δ(uδ, v) = lim +δ→0 Bρ,δ(uδ − u, v) + lim +δ→0 Bρ,δ(u, v) = +ˆ +Ω +∇u · ∇v dx . +Therefore u satisfies (7.5), and so in summary u is the unique weak solution in H1(Ω) of the +Poisson equation (1.15) with Dirichlet boundary conditions (1.7). +□ + +NONLOCAL BOUNDARY-VALUE PROBLEMS +41 +7.3.1. Regularizing rough data. The next theorem gives an explicit construction of the mollified +sequence fδ that satisfies (7.1)-(7.2). +Theorem 7.6. Let Ω ⊂ Rd satisfy (AΩ) and ρ satisfy (A1)-(A2). For f ∈ H−1(Ω) and some +α ≥ 0 fixed, define the distribution +(7.6) +fδ := K∗ +δ,αf . +Then fδ satisfies (7.1)-(7.2). +Proof. Similarly to the proof of (6.20), it is a consequence of Lemma 6.5 that +T[Kδ,αϕ] ≡ 0 , ∀ϕ ∈ C∞ +c (Ω) . +Thanks to the continuity of the operator Kδ,α : Wδ,2(Ω) → H1(Ω) implied by (6.6)-(6.8), and +thanks to continuity in Wδ,2(Ω) of the trace operator, we conclude that Kδ,αϕ ∈ H1 +0(Ω) whenever +ϕ ∈ Wδ,2 +0 (Ω). Hence +| +� +ϕ, K∗ +δ,αf +� +Wδ,2 +0 (Ω),[Wδ,2(Ω)]∗ | = | ⟨f, Kδ,αϕ⟩H1 +0(Ω),H−1(Ω) | ≤ C ∥f∥H−1(Ω) ∥Kδ,αϕ∥H1(Ω) , +and so by (6.6)-(6.8) we conclude that +∥fδ∥[Wδ,2 +0 (Ω)]∗ = +��K∗ +δ,αf +�� +[Wδ,2 +0 (Ω)]∗ ≤ C ∥f∥H−1(Ω) . +The estimate +∥ηδfδ∥L2(Ω) + +��η2 +δ∇fδ +�� +L2(Ω) ≤ C ∥f∥H−1(Ω) +follows from (6.18), and therefore (7.2) is established by the previous two estimates. +To prove (7.1), we first note that by (6.26) +(7.7) +lim +δ→0 ∥Kδ,αϕ − ϕ∥H1(Ω) = 0 , +∀ϕ ∈ C∞ +c (Ω) . +Now let ϕ ∈ H1 +0(Ω), and let (ϕn) be a sequence in C∞ +c (Ω) that converges to ϕ in H1(Ω). Then +by (6.20) +| ⟨fδ − f, ϕ⟩ | ≤ | ⟨fδ − f, ϕn⟩ | + C(ρ, Ω) ∥f∥H−1(Ω) ∥ϕn − ϕ∥H1(Ω) += | ⟨f, Kδ,αϕn − ϕn⟩ | + C(ρ, Ω) ∥f∥H−1(Ω) ∥ϕn − ϕ∥H1(Ω) +≤ C(ρ, Ω) ∥f∥H−1(Ω) +� +∥Kδ,αϕn − ϕn∥ + ∥ϕn − ϕ∥H1(Ω) +� +. +The second quantity can be made as small as desired by fixing n large, and so using (7.7) upon +taking δ → 0 gives +lim sup +δ→0 +| ⟨fδ − f, ϕ⟩ | < ε for any ε > 0 . +Thus (7.1) is proved. +□ +7.3.2. Convergence of normal derivatives. Since we know uδ, u ∈ H1(Ω), we also know that ∂uδ +∂ν +and ∂u +∂ν are well-defined distributions in H− 1 +2 (∂Ω). +Theorem 7.7. Let Ω ⊂ Rd satisfy (AΩ). Suppose that f ∈ L2(Ω). Let uδ be the solution of +(7.3) with Dirichlet boundary condition (1.7) and Poisson data fδ, where fδ is defined in (7.6), +and Dirichlet data g ∈ H +1 +2 (∂Ω). Then ∂uδ +∂ν ∈ H− 1 +2(∂Ω) with the uniform estimate +���� +∂uδ +∂ν +���� +H− 1 +2 (∂Ω) +≤ C(d, ρ, Ω) +� +∥f∥L2(Ω) + ∥g∥H +1 +2 (∂Ω) +� +. + +42 +JAMES M. SCOTT AND QIANG DU +Proof. It is clear from (6.19) that +∥fδ∥L2(Ω) ≤ C(d, ρ, Ω) ∥f∥L2(Ω) . +Since (1.6) holds and since fδ ∈ L2(Ω), we can now test against a wider class of functions to get +ˆ +Ω +Lδu(x)ϕ(x) dx = +ˆ +Ω +fδ(x)ϕ(x) dx , +∀ϕ ∈ C∞(Ω) . +Let v ∈ H1/2(∂Ω), and let ¯v ∈ H1(Ω) be any extension to Ω. Then by the nonlocal Green’s +identity (1.5) (See case ii) of Proposition 4.5) +�∂uδ +∂ν , v +� += Bρ,δ(uδ, v) − +ˆ +Ω +Lδuδv dx += Bρ,δ(uδ, v) − +ˆ +Ω +fδv dx . +Therefore using the energy estimate (5.6), the embedding Theorem 1.5, and boundedness of the +extension ¯v +���� +�∂uδ +∂ν , v +����� ≤ [uδ]Wδ,2(Ω)[v]Wδ,2(Ω) + ∥f∥L2(Ω) ∥v∥L2(Ω) +≤ C +� +∥f∥L2(Ω) + ∥g∥H +1 +2 (∂Ω) +� +∥v∥H +1 +2 (∂Ω) , +which gives the result. +□ +Theorem 7.8. Let Ω ⊂ Rd satisfy (AΩ). Suppose that f ∈ L2(Ω). Let uδ be a solution of +(7.3) with Dirichlet data g and Poisson data fδ, where fδ is defined in (7.6), and Dirichlet +data g ∈ H +3 +2(∂Ω). Let u be the unique variational solution of the Poisson equation (1.15) and +Dirichlet boundary condition (1.7) with Poisson data f and Dirichlet data g. Then +lim +δ→0 +�∂uδ +∂ν , v +� += +� ∂u +∂ν , v +� +, +∀v ∈ H +1 +2 (∂Ω) . +Proof. Note that the mollification fδ is defined pointwise, that is, fδ(x) = K∗ +δ,αf(x). In addition, +by (6.19) fδ ∈ L2(Ω), with +∥fδ∥L2(Ω) ≤ C(ρ, Ω) ∥f∥L2(Ω) . +We have from (7.4) that uδ ∈ H1(Ω) with +∥uδ∥H1(Ω) ≤ C(d, ρ, Ω) +� +∥f∥H−1(Ω) + ∥g∥H +1 +2 (∂Ω) +� += C +� +∥f∥L2(Ω) + ∥g∥H +1 +2 (∂Ω) +� +. +Now, (1.6) holds, hence Lδuδ ∈ L2(Ω) since fδ ∈ L2(Ω), and we can test against a wider class +of functions to get +ˆ +Ω +Lδu(x)ϕ(x) dx = +ˆ +Ω +fδ(x)ϕ(x) dx , +∀ϕ ∈ C∞(Ω) . +Let v ∈ H1/2(∂Ω), and let ¯v ∈ H1(Ω) be any extension of v to Ω. Then by the nonlocal +Green’s identity (1.5) (See case ii) of Proposition 4.5) +�∂uδ +∂ν , v +� += Bρ,δ(uδ, ¯v) − +ˆ +Ω +Lδuδ(x)¯v(x) dx += Bρ,δ(uδ, ¯v) − +ˆ +Ω +fδ(x)¯v(x) dx . +Now, let u ∈ H2(Ω) be the unique variational solution to (1.15) and (1.7) with Poisson +data f and Dirichlet data g (global H2 regularity of u for f ∈ L2(Ω) and g ∈ H +3 +2 (∂Ω) is proved + +NONLOCAL BOUNDARY-VALUE PROBLEMS +43 +in [37, Theorem 4.14]). Then uδ converges to u weakly in H1(Ω) by Theorem 7.5. From the +nonlocal Green’s identity (1.5) (See case ii) of Proposition 4.5) +�∂u +∂ν , v +� += Bρ,δ(u, ¯v) − +ˆ +Ω +Lδu(x)¯v(x) dx += Bρ,δ(u, ¯v) + +ˆ +Ω +((−∆u(x)) − Lδu(x))¯v(x) dx − +ˆ +Ω +−∆u(x)¯v(x) dx += Bρ,δ(u, ¯v) + +ˆ +Ω +((−∆u(x)) − Lδu(x))¯v(x) dx − +ˆ +Ω +f(x)¯v(x) dx . +Therefore, we have +�∂uδ +∂ν , v +� +− +�∂u +∂ν , v +� += Bρ,δ(uδ − u, ¯v) + +ˆ +Ω +(Lδu(x) − (−∆u(x)))¯v(x) dx +− +ˆ +Ω +(fδ(x) − f(x))¯v(x) dx . +The result then follows from Theorem 7.3, Theorem 7.1, and (6.23). +□ +Remark 7.9. The assumption g ∈ H +3 +2(∂Ω) was made in order to easily obtain the strong L2 +convergence of Lδu to −∆u. The case g ∈ H +1 +2(∂Ω) can likely be treated by introducing the +solution space {u ∈ L2(Ω) : ∆u ∈ L2(Ω)} for the classical problem and then redoing the +convergence proofs of Section 7.1 and Section 7.2 more carefully; this will be investigated in +future works. +7.4. The Neumann problem. We now study the Neumann problem with δ varying, and +analyze the case δ → 0. +Theorem 7.10. Let Ω ⊂ Rd satisfy (AΩ). Let f ∈ [H1(Ω)]∗ and g ∈ H− 1 +2 (∂Ω) with ⟨f, 1⟩ + +⟨g, 1⟩ = 0. Suppose for each δ ∈ (0, δ0) there exists a function fδ ∈ L2 +loc(Ω) that satisfies +(7.8) +⟨fδ, 1⟩ = ⟨f, 1⟩ , +∀δ > 0 +with +(7.9) +lim +δ→0+ ⟨fδ − f, v⟩[H1(Ω)]∗,H1(Ω) = 0 , +∀v ∈ H1(Ω) , +and that +(7.10) +� +∥fδ∥[Wδ,2(Ω)]∗ + ∥ηδfδ∥L2(Ω) + +��η2 +δ∇fδ +�� +L2(Ω) +� +≤ C0 ∥f∥[H1(Ω)]∗ . +Then the corresponding solutions uδ of (5.9) with Poisson data fδ and Neumann data g satisfy +(7.11) +∥uδ∥H1(Ω) ≤ C(Ω, ρ) +� +C0 ∥f∥[H1(Ω)]∗ + ∥g∥H− 1 +2 (∂Ω) +� +. +Proof. The result follows from (6.38) and (7.10). +□ +Theorem 7.11. Let Ω ⊂ Rd satisfy (AΩ). Suppose that f ∈ [H1(Ω)]∗ and g ∈ H− 1 +2 (∂Ω) with +⟨f, 1⟩ + ⟨g, 1⟩ = 0, and let fδ be a sequence of functions that satisfy (7.8)-(7.9)-(7.10). Then the +solutions uδ ∈ ˚ +H1(Ω) to (5.9) converge weakly in H1(Ω) to a function u ∈ ˚ +H1(Ω), where u is the +variational solution to the Poisson equation with inhomogeneous Neumann boundary conditions +(1.15)-(1.8). +Proof. Since the solutions satisfy the uniform H1 bound (7.11), it follows that they converge +weakly in H1(Ω) to a function u. Since u is the weak limit of the uδ, we have by Lemma 6.9 +¯ρ +ˆ +Ω +u(x) dx = lim +δ→0 +ˆ +Ω +Φδ(x)uδ(x) dx = 0 +and so u ∈ ˚ +H1(Ω). + +44 +JAMES M. SCOTT AND QIANG DU +We now prove that u is the unique weak solution in ˚ +H1(Ω) of the Poisson equation, i.e. +ˆ +Ω +∇u · ∇v dx = ⟨f, v⟩ + ⟨g, Tv⟩ , +∀v ∈ ˚ +H1(Ω) . +Let v ∈ ˚ +H1(Ω) be arbitrary. Then v − (Φδv)Ω can be used as a test function in (5.9), and so for +each δ > 0 +⟨fδ, v⟩ + ⟨g, Tv⟩ = ⟨fδ, v − (Φδv)Ω⟩ + ⟨g, T(v − (Φδv)Ω)⟩ = Bρ,δ(uδ, v − (Φδv)Ω) = Bρ,δ(uδ, v) . +On one hand, (7.9) implies +lim +δ→0 ⟨fδ, v⟩ = ⟨f, v⟩ , +and on the other hand, Theorem 7.3 and Theorem 7.2 imply that +lim +δ→0 Bρ,δ(uδ, v) = lim +δ→0 Bρ,δ(uδ − u, v) + lim +δ→0 Bρ,δ(u, v) = +ˆ +Ω +∇u · ∇v dx . +Therefore u is the unique weak solution in ˚ +H1(Ω) of the Poisson equation (1.15)-(1.8). +□ +7.4.1. Regularizing rough data. The next theorem gives an explicit construction of the mollified +sequence fδ that satisfies (7.8)-(7.9)-(7.10) for a subset of [H1(Ω)]∗. The subset is defined as +follows: for f0 ∈ L2(Ω) and f1 ∈ L2(Ω; Rd) and supp |f1| ⋐ Ω, define a distribution f by +(7.12) +⟨f, v⟩ := +ˆ +Ω +f0(x)v(x) dx + +ˆ +Ω +f1(x) · ∇v(x) dx +∀v ∈ H1(Ω) , +and define H∗ to be the subset of [H1(Ω)]∗ given by all such pairs (f0, f1), i.e. +H∗ := +� +f ∈ [H1(Ω)]∗ : f given by (7.12) and supp |f1| ⋐ Ω +� +. +Theorem 7.12. For f ∈ H∗ given via f0 ∈ L2(Ω) and f1 ∈ L2(Ω; Rd), define the distribution +fδ ∈ [H1(Ω)]∗ by +(7.13) +⟨fδ, v⟩ = +ˆ +Ω +(f δ +0(x) + F δ +1 (x))v(x) dx , +∀v ∈ H1(Ω) , +where +f δ +0(x) := K∗ +δ,αf0(x) +and +F δ +1 (x) := +ˆ +Ω +∇yρδ,α(x, y) · +f1(y) +Φδ,α(y) dy − +ˆ +Ω +ρδ,α(x, y)∇yΦδ,α(y) +Φδ,α(y)2 +· f1(y) dy . +Then fδ satisfies (7.8)-(7.9)-(7.10). +Proof. Using the same argument as in the proof of Lemma 6.5, we see that F δ +1 has compact +support in Ω, so therefore by Lemma 2.3, Theorem 2.6, (2.16) and (2.19) +���F δ +1 +��� +L2(Ω) ≤ +C(ρ, Ω) +dist(supp F δ +1 , Ω) ∥f1∥L2(Ω) . +Thus the expression (7.13) is an absolutely convergent integral for any v ∈ H1(Ω). +To see (7.8) use the definitions of Kδ,α and Φδ,α and the symmetry of ρδ,α(x, y): +ˆ +Ω +f δ +0(x) dx = +ˆ +Ω +�ˆ +Ω +ρδ,α(x, y)dx +� f0(y) +Φδ,α(y) dy = +ˆ +Ω +f0(x) dx = 0 , +ˆ +Ω +F δ +1 (x) dx = +ˆ +Ω +�ˆ +Ω +∇yρδ,α(x, y)dx +� +· +f1(y) +Φδ,α(y) dy dy +− +ˆ +Ω +∇yΦδ,α(y) +Φδ,α(y) +· f1(y) dy . + +NONLOCAL BOUNDARY-VALUE PROBLEMS +45 +Now we prove (7.10). First, by (6.18) +(7.14) +���f δ +0 +��� +[Wδ,2(Ω)]∗ + +���ηδf δ +0 +��� +L2(Ω) + +���η2 +δ∇f δ +0 +��� +L2(Ω) ≤ C ∥f0∥L2(Ω) . +Second, by Lemma 2.3, Theorem 2.6, (2.16) and (2.19) +(7.15) +���ηδF δ +1 +��� +L2(Ω) + +���η2 +δ∇F δ +1 +��� +L2(Ω) ≤ C(ρ, Ω) ∥f1∥L2(Ω) . +Third, for any v ∈ C∞(Ω) the definitions of Kδ,α and Φδ,α and the symmetry of ρδ,α gives +ˆ +Ω +F δ +1 (x)v(x) dx = +ˆ +Ω +1 +Φδ,α(y) +�ˆ +Ω +∇yρδ,α(x, y)v(x) dx +� +f1(y) dy +− +ˆ +Ω +∇yΦδ,α(y) +Φδ,α(y)2 +�ˆ +Ω +ρδ,α(x, y)v(x) dx +� +f1(y) dy += +ˆ +Ω +∇y [Kδ,αv(y)] · f1(y) dy ; +(7.16) +Note that interchanging the integrals is justified since f1 is compactly supported. Therefore by +Corollary 6.3 +(7.17) +��� +� +F δ +1 , v +���� ≤ C(ρ, Ω) ∥f1∥L2(Ω) ∥v∥Wδ,2(Ω) , +∀v ∈ Wδ,2(Ω) . +Then (7.10) follows from (7.14)-(7.15)-(7.17). +Now let v ∈ H1(Ω) and let K = supp |f1|, which is compactly contained in Ω by assumption. +Then by (7.16) +⟨fδ, v⟩ = +ˆ +Ω +f δ +0(x)v(x) + F δ +1 (x)v(x) dx += +ˆ +Ω +K∗ +δ,αf0(x) v(x) dx + +ˆ +K +∇ [Kδ,αv(x)] · f1(x) dx , +and so by Hölder’s inequality +| ⟨fδ − f, v⟩ | ≤ +��K∗ +δ,αf0 − f0 +�� +L2(Ω) ∥v∥L2(Ω) + ∥∇Kδ,αv − ∇v∥L2(K) ∥f1∥L2(Ω) . +Thus (7.9) follows from (6.23) and (6.26). +□ +8. Conclusion +In this work we presented a systematic treatment of varational problems for nonlocal op- +erators with classical boundary conditions. These models are potentially useful for applications +incorporating long-range phenomena in which classical, local boundary conditions are more nat- +ural or preferable. They may offer sound alternatives to some ad hoc approaches that might +produce nonphysical or undesirable artifacts. Meanwhile, we developed a series of mathematical +tools to analyze the nonlocal problems involving heterogeneous localizations and offered new +analytical insight. A nonlocal Green’s identity was established, which is a key tool for the study +here, and will likely be useful in the treatment of other types of nonlocal boundary-value prob- +lems as well, such as Robin or oblique boundary conditions. It also paves a path to study issues +like Dirichlet-to-Neumann maps in the nonlocal context, and motivates further extensions to +time-dependent problems and nonlinear problems. +Concerning the assumptions made in the current work, we first remark on the choice of +superlinear localization at ∂Ω. Instead, if the choice ηδ(x) ≈ min{δ, dist(x, ∂Ω)} is made, then +the constant-to-variable transition layer is necessarily of the same width. The optimal estimate +for derivatives is +|Dβηδ(x)| ≤ Cηδ(x)1−|β| . + +46 +JAMES M. SCOTT AND QIANG DU +As a result, Lδ does not satisfy an operator bound uniform in δ, and it is not “well-matched” +with its local counterpart −∆. This is made precise in a statement of a type of Green’s identity +for such a heterogeneous localization. +Proposition 8.1. Let Ω ⊂ Rd be a bounded C2 domain. +Then there exists a function ηδ +satisfying the following: +i) ηδ(x) = dist(x, ∂Ω) for all x ∈ Ω with dist(x, ∂Ω) < κ0δ, +ii) ηδ(x) = δ for all x ∈ Ω with dist(x, ∂Ω) > +δ +κ0, +iii) ηδ ∈ C2(Ω) with |∇ηδ(x)| ≤ κ1 and |∇2ηδ(x)| ≤ κ2 +δ +for all x ∈ Ω, where κ1, κ2 are +positive constants depending at most on κ0, d and Ω. +iv) There is a positive constant ¯κ0 depending only on at most κ0, d and Ω such that ηδ(x) ≤ +¯κ0 min{δ, dist(x, ∂Ω)} for all x ∈ Ω. +Suppose ρ satisfies (A1)-(A2). Then for any u, v ∈ C2(Ω) +Bρ,δ(u, v) = +ˆ +Ω +Lδu(x) · v(x) dx + Cρ +ˆ +∂Ω +∂u +∂ν (x) · v(x) dσ(x) , +where Cρ is a constant defined by +Cρ := +ˆ +B(0,R0)∩{zd>0} +zd ln +�1 + zd +1 − zd +� +ρ(|z|)dz > 1 . +Immediately we can see one difficulty: some of the boundary information is still carried +within the term Lδu. Indeed, +lim +δ→0 Bρ,δ(u, v) = +ˆ +Ω +∇u · ∇v dx , +while +lim +δ→0 Lδu(x) = (1 − Cρ)∂u +∂ν (x)H d−1⌞∂Ω − ∆u(x) +in the sense of measures. That is, +lim +δ→0 +ˆ +Ω +Lδu(x) · v(x) dx = − +ˆ +Ω +∆u(x)v(x) dx + (1 − Cρ) +ˆ +∂Ω +∂u +∂ν (x) · v(x) dσ(x) . +We also remark on the possibility of proving (1.5) for domains Ω that are not C2. Such a +result is desirable in applications, for example applications in continuum mechanics that polyg- +onal domains. If ηδ is similarly defined for a C1,β domain for 0 < β < 1 then ηδ is only C1,β, and +so then Lδu does not define a function in L1(Ω) even if u ∈ C∞(Ω). A more suitable definition +of ηδ for domains with less regularity remains to be investigated. For bounded Lipschitz domain +Ω, the function ηδ is C1 in Ω but not C2. Thus, the heterogeneous localization parameter should +be replaced with a smooth approximation comparable to dist(x, ∂Ω)2 near ∂Ω and satisfying +the appropriate estimates (indeed such functions exist; see [56]). At the same time, it seems +necessary to assure that the regularization would not violate condition i) in order to prove the +nonlocal Green’s identity. We leave such a systematic investigation for a future work. +Lastly, there are many other interesting open questions left. For example, although our +focus here is on rough data, one may also study problems with more regular data and how they +further impact the solution regularity. As another example, it is also natural to ask whether +the current study can be extended to more general variational problems. On the latter, let us +point out that the weak formulations of the nonlocal problems with local boundary conditions +can also be formulated as energy minimization problems. For instance, for (5.1), we have an +equivalent form +(8.1) +min +u∈Wδ,2 +0 (Ω) +Bρ,δ(u, u) − ⟨f, u⟩ , + +NONLOCAL BOUNDARY-VALUE PROBLEMS +47 +for the nonlocal problem with homogeneous Dirichlet boundary conditions. In the spirit of (8.1), +via direct methods of the calculus of variations, we can further extend the earlier discussions on +the linear nonlocal problems to certain nonlinear cases as well. While more extensive studies +will be left to future works, we state the following as an illustration. +Theorem 8.2. Let f ∈ H−1(Ω), and let fδ be a sequence satisfying (7.1)-(7.2). Define 2∗ := +2d +d−2, 2∗ := (2∗)′ = +2d +d+2, and fix p ∈ [2, 2∗). Then there exists a unique uδ ∈ Wδ,2(Ω) ∩ Lp(Ω) +that satisfies +uδ = +argmin +v∈Wδ,2 +0 (Ω)∩Lp(Ω) +Bρ,δ(v, v) + 1 +p +ˆ +Ω +|K∗ +δ,αv(x)|p dx − ⟨fδ, v⟩ . +Further, uδ ∈ H1 +0(Ω). Moreover, as δ → 0, {uδ} is uniformly bounded in H1 +0(Ω) and so has a +weakly convergent subsequence in H1(Ω). The weak limit u satisfies +(8.2) +u = argmin +v∈H1 +0(Ω) +ˆ +Ω +|∇v(x)|2 dx + 1 +p +ˆ +Ω +|v(x)|p dx − ⟨f, v⟩ . +Proof. First, using direct methods of the calculus of variations one can show that there exists a +unique minimizer uδ ∈ Wδ,2 +0 (Ω) ∩ Lp(Ω) since the nonlocal functional is coercive, weakly lower +semicontinuous, and convex with strongly convex principal part. +Next, we have an energy estimate; using the Poincaré inequality Theorem 4.7 +[uδ]2 +Wδ,2(Ω) + 1 +p +��K∗ +δ,αuδ +��p +Lp(Ω) ≤ ∥fδ∥[Wδ,2(Ω)]∗ ∥uδ∥Wδ,2(Ω) +≤ C ∥fδ∥[Wδ,2(Ω)]∗ +� +[uδ]2 +Wδ,2(Ω) + 1 +p +��K∗ +δ,αuδ +��p +Lp(Ω) +�1/2 +. +Then the uniform bound +[uδ]2 +Wδ,2(Ω) + 1 +p +��K∗ +δ,αuδ +��p +Lp(Ω) ≤ C ∥fδ∥[Wδ,2(Ω)]∗ ≤ C ∥f∥2 +H−1(Ω) +follows from (7.2). +Now, uδ satisfies the Euler-Lagrange equation +Lδuδ(x) = fδ(x) − Kδ,α[|K∗ +δ,αuδ|p−2K∗ +δ,αuδ](x) +almost everywhere in Ω. Then we have +uδ(x) = Kδ,2uδ(x) + +fδ(x) +2Φδ,2(x) − +Kδ,α[|K∗ +δ,αuδ|p−2K∗ +δ,αuδ](x) +2Φδ,2(x) +. +By (6.6), (6.7), and (7.2), +∥uδ∥H1(Ω) ≤ C ∥u∥Wδ,2(Ω) + C ∥f∥H−1(Ω) + C +��|K∗ +δ,αuδ|p−2K∗ +δ,αuδ +�� +H−1(Ω) +≤ C ∥f∥H−1(Ω) + C +��|K∗ +δ,αuδ|p−2K∗ +δ,αuδ +�� +H−1(Ω) . +Now, for any function ϕ ∈ Lp(Ω) for p ∈ [2, 2∗), one can use Hölder’s inequality and Sobolev +embedding to obtain +��ϕp−1�� +H−1(Ω) ≤ +��ϕp−1�� +L2∗(Ω) ≤ C(Ω, p) ∥ϕ∥Lp(Ω) . +Therefore by using (6.19) adapted for general Lebesgue spaces and the energy estimate +��|K∗ +δ,αuδ|p−2K∗ +δ,αuδ +�� +H−1(Ω) = +��K∗ +δ,αuδ +�� +Lp(Ω) ≤ C ∥f∥H−1(Ω) . +Then the uniform bound follows. Hence a subsequence of uδ converges weakly in H1(Ω) to a +function u. Using standard techniques from direct methods it is not difficult to show that u +satisfies (8.2) thanks to the weak lower semicontinuity of the functionals as well as the continuity +properties of Bρ,δ and Kδ,α. +□ + +48 +JAMES M. 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SCOTT AND QIANG DU +[57] Yunzhe Tao, Xiaochuan Tian, and Qiang Du. Nonlocal models with heterogeneous localization and their +application to seamless local-nonlocal coupling. Multiscale Modeling & Simulation, 17(3):1052–1075, 2019. +[58] Xiaochuan Tian and Qiang Du. Trace theorems for some nonlocal function spaces with heterogeneous local- +ization. SIAM Journal on Mathematical Analysis, 49(2):1621–1644, 2017. +[59] Enrico Valdinoci. From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat. Apl. +SeMA, 49:33–44, 2009. +Department of Applied Physics and Applied Mathematics, Columbia University, New York, +NY 10027 +Email address: jms2555@columbia.edu +Department of Applied Physics and Applied Mathematics, and the Data Science Institute, +Columbia University, New York, NY 10027 +Email address: qd2125@columbia.edu + diff --git a/QdE1T4oBgHgl3EQfHQMq/content/tmp_files/load_file.txt b/QdE1T4oBgHgl3EQfHQMq/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..92792e0734fc703d74cede54463fa7a488810e20 --- /dev/null +++ b/QdE1T4oBgHgl3EQfHQMq/content/tmp_files/load_file.txt @@ -0,0 +1,2002 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf,len=2001 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content='02923v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content='AP] 7 Jan 2023 NONLOCAL BOUNDARY-VALUE PROBLEMS WITH LOCAL BOUNDARY CONDITIONS JAMES M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' SCOTT AND QIANG DU Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' We describe and analyze nonlocal integro-differential equations with classical local boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' The interaction kernel of the nonlocal operator has horizon parameter dependent on position in the domain, and vanishes as the boundary of the domain is approached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' This heterogeneous localization allows for boundary values to be captured in the trace sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' We state and prove a nonlocal Green’s identity for these nonlocal operators that involve local boundary terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' We use this identity to state and establish the well-posedness of variational formulations of the nonlocal problems with several types of classical boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' We show the consistency of these nonlocal boundary-value problems with their classical local counterparts in the vanishing horizon limit via the convergence of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' The Poisson data for the local boundary-value problem is permitted to be quite irregular, belonging to the dual of the classical Sobolev space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' Heterogeneously mollifying this Poisson data for the local problem on the same length scale as the horizon and using the regularity of the interaction kernel, we show that the solutions to the nonlocal boundary-value problem with the mollified Poisson data actually belong to the classical Sobolev space, and converge weakly to the unique variational solution of the classical Poisson problem with original Poisson data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' Introduction There has been much recent interest in nonlocal problems on a bounded domain Ω ⊂ Rd: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content='1) Lu = f in Ω , associated to a nonlocal integral operator (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content='2) Lu(x) := 2 ˆ Rd γ(x, y)(u(x) − u(y)) dy , defined for measurable functions u : Rd → R and a nonlocal, symmetric and (often) nonnegative kernel γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' These operators appear widely in both analysis and applications [2,4,5,8,9,13–15,17, 19,23,24,32,39,40,45,48,49,51,53,59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' Here, symmetry means γ(x, y) = γ(y, x), so that we can identify L with a variational form associated with a quadratic nonlocal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' Earlier studies of variational formulations of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content='1) on bounded domains have taken several different paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' Along the path that γ = γ(x, y) has a singularity at the diagonal x = y, both volume- constraint problems and classical boundary-value problems have been investigated, see for ex- ample [3,18,30,52] and additional references cited therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' If L defines a hypersingular integral operator, in particular, then classical boundary values can be prescribed via the trace operator, see [1, 50] for the cases of singular kernels that give rise to solutions in fractional Sobolev- Slobodeckij spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' Down another path, γ = γ(x, y) is a compactly supported and translation invariant kernel, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=', γ(x, y) = δ−d−2ω(|x − y|/δ) for a function ω supported in the unit interval (0, 1) and a constant (horizon parameter δ) that measures the range of nonlocal interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' One natural route to take is to define the so-called nonlocal volumetric constraint to complement the equation 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' 45K05, 35J20, 46E35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' nonlocal equations, boundary-value problems, Poisson problem, Green’s identity, nonlocal operators, heterogeneous localization, vanishing horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' This work is supported in part by the NSF DMS-1937254 and DMS-2012562.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' 1 2 JAMES M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' SCOTT AND QIANG DU defined on Ω [24,25,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' An example is the prescription of u(x) in a layer consisting of x ∈ Ωc with dist(x, Ω) < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' An alternative is to modify the nonlocal interaction rules involving u = u(x) in a layered domain, say, for x ∈ Ω with dist(x, ∂Ω) < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' These volumetric conditions can recover traditional boundary conditions in the local limit as δ → 0 under suitable conditions, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=', [6, 22, 28, 35, 44, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' Meanwhile, in the δ → ∞ limit with a rescaled fractional kernel, these problems are related to studies of fractional differential equations defined on a bounded domain [7,20,33,41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' In addition, one can find connections to the continuum limits of discrete graph operators and discrete particle interactions [8,17,36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' For various nonlocal problems, studies of their well-posedness subject to nonlocal volumetric constraints can be found, for example, in [25,46], which offered desirable mathematical insight as demonstrated for a number of applications such as the peridynamics models developed in mechanics [16,29,43,54,55], nonlocal diffusion and jump processes [10,24] and nonlocal Stokes equations for the analysis of smoothed particle hydrodynamics [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' Still another path is to mix classical boundary conditions and volume-constraint conditions in constitutive models that blend local and nonlocal models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' For an extensive discussion relating to the many choices of blended models in applications such as peridynamics, see the survey [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' We are interested in boundary-value problems for nonlocal problems on a bounded domain in the classical sense, that is, the boundary conditions are prescribed on ∂Ω only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' The motiva- tion is two-fold: first, while the nonlocal constraints are natural, they are not perfect choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' Theoretically, nonlocal constraints may raise unintended concerns about the regularity of so- lutions, for instance, non-constant functions vanishing in a layer of nonzero measure no longer enjoy analyticity, and solutions of problems with smooth kernels may experience non-physical or undesirable jumps at the boundary due to unmatched nonlocal constraints [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' In practice, developers of simulation codes for applications of nonlocal models, have ample practical reasons to keep local boundary conditions in implementation even though a nonlocal model might be derived and/or deemed a better modeling choice in the domain of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' To allow for the prescription of local boundary conditions, the nonlocal operator L and the nonlocal solution spaces must be defined so that boundary values of the solutions make sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' For a nonlocal operator associated with a kernel γ = γ(x, y) that does not have sufficient singularity on the diagonal x = y, it means that some localizing property near the boundary should hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' For instance, in [26,57,58], with a pair of constants β ∈ (0, 1) and δ > 0, a function hδ(x) = min{δ, β dist(x, ∂Ω)} is introduced to characterize the extent of nonlocal interactions at a point x ∈ Ω, instead of taking a constant δ as the horizon parameter everywhere in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE1T4oBgHgl3EQfHQMq/content/2301.02923v1.pdf'} +page_content=' A kernel similar to γ(x, y) = 1 2hδ(x)d+2 1{|x−y| 0, +Hξ(x) := +� +� +� +� +� +exp +� +−(1 + ξx)−1/ξ� +if ξ ̸= 0 +exp{− exp{−x}} +if ξ = 0 +denotes the GEV distribution (Embrechts et al., 1997). +Assumption 3.1. There exists z < xF such that on the interval (z, xF), F is twice differ- +entiable and f is positive, decreasing and convex. +Assumption 3.2. F belongs to the maximum domain of attraction (MDA) of Hξ, i.e., +F ∈ MDA(Hξ). In another words, there exist normalization constants cn > 0, dn ∈ R such +that c−1 +n (Mn − dn) ⇒ Hξ for some ξ ∈ R as n → ∞ where Mn := max(X1, · · · , Xn) is the +sample maxima and X1, · · · , Xn are random realizations of X. +We recall that there are three different types of GEV distribution: the Fr´echet dis- +tribution φα with ξ = α−1 > 0, the Gumbel distribution Λ with ξ = 0 and the Weibull +distribution ψα with ξ = −α−1 < 0. ξ is a shape parameter to govern the tail behavior of +the distribution. Indeed, according to Embrechts et al. (1997), in the Fr´echet case, xF = ∞ +and limx→∞ +¯F(tx) +¯F(x) = t− 1 +ξ , t > 0. Such an aymptotic behavior is also known as regularly vary- +ing at ∞ with index −1/ξ, denoted for short by ¯F ∈ RV−1/ξ. The larger is ξ, the more +slowly the tail of ¯F decays, and hence the heavier the tail of F is. Similarly, when ξ = 0 +with xF = ∞ (Gumbel distribution), the tail is lighter than those in the Fr´echet case. +In the case that ξ = 0, we make an additional assumption that is broadly satisfied by +the textbook distributions such as normal, Gamma and exponential distributions. Note +that Assumption 3.3 together with the above assumptions with ξ = 0 imply that F is a +von Mises function. +Assumption 3.3. limx↑xF +F(x)f′(x) +f2(x) += −1. +It is known from Embrechts et al. (1997) that if ξ > 0, then xF = ∞; if ξ < 0, then +18 + +xF < ∞; if ξ = 0, then xF can be either finite or infinite. In fact, under our assumptions, +we can focus on the case that xF = ∞, which is justified by the following two propositions: +Proposition 3.1. Suppose that F satisfies Assumption 3.1 and Assumption 3.2 with ξ < 0. +Let Y = 1/(xF−X). Use FY and fY to denote the distribution function and density function +of Y . Then there exists zY < ∞ such that on (zY , ∞), FY is twice differentiable and fY is +positive, decreasing and convex. Moreover, FY ∈ MDA(H−ξ). +Proposition 3.2. Suppose that F satisfies Assumption 3.1, Assumption 3.2 with ξ = 0 and +additionally, Assumption 3.3. Moreover, we suppose that xF < ∞. Let Y = 1/(xF − X). +Use FY and fY to denote the distribution function and density function of Y . Then there +exists zY < ∞ such that on (zY , ∞), FY is twice differentiable and fY is positive, decreasing +and convex. Moreover, FY ∈ MDA(Λ) with infinite right endpoint. In addition, we have +that +lim +x→∞ +¯FY (x)f ′ +Y (x) +f 2 +Y (x) += −1. +(7) +By the above two propositions, under our assumptions, if xF < ∞, then we may define +Y = 1/(xF − X), which also satisfies our assumptions. Knowing F(x), x ≤ a implies that +we know FY , the distribution function of Y , up to 1/(xF −a). Therefore, we may transform +the problem into an equivalent one with infinite right endpoint. From now on, we assume +that xF = ∞ without loss of generality and either F ∈ MDA(Hξ) for some ξ > 0, or +F ∈ MDA(Λ). +First we consider estimating tail probabilities ψ(P) = E[h(X)] with h(x) = I(x > b), +i.e. ψ(P) = P(X > b) = +� ∞ +b +f(x)dx, where b = b(a) satisfies that a ≤ b < ∞. We extract +three constants β, η, ν from the known information f(x), x ≤ a: +β = β(a) := 1 − F(a) = ¯F(a); η = η(a) := f(a); ν = ν(a) := −f ′ ++(a). +19 + +For simplicity of further use, we define +µ = µ(a) := η +ν , σ = σ(a) := 2β +ν . +For β, η, ν > 0, we consider the RO problem formulated in the following form: +max +f +� ∞ +a +I(x > b)f(x)dx +s.t. +� ∞ +a +f(x)dx = β, +f(a) = f(a+) = η, +f ′ ++(a) ≥ −ν, +f convex for x ≥ a, +f(x) ≥ 0 for x ≥ a. +(8) +We note that (8) is equivalent to P(P2 +a,η,η,ν, I(x > a), {β}), so we are indeed considering a +special case of the previously defined framework. From now on, the optimal value to the +above problem is denoted by z∗ = z∗(a, b). By applying the results in Lam and Mottet +(2017), we get the following theorem: +Theorem 3.1. If η2 < 2βν, then the optimal value of (8), denoted by z∗, is +z∗ = +� +� +� +� +� +� +� +� +� +ν +2(σ − µ2) +if µ ≤ b − a; +ν +2[σ − 2(b − a)µ + (b − a)2] +if µ > b − a. +(9) +To get some intuition on Theorem 3.1, note that if we draw a line from (a, η) with slope +−ν, then it hits 0 at (a + µ, 0). The tail extrapolation of any feasible density function +must be above this straight line. On the other hand, they can be as close as possible. +Thus intuitively z∗ is equal to β subtracted by the area of the shaded region in Figure 1, +which exactly coincides with the results in the theorem. To quantify the conservativeness +in estimating tail probabilities, we need to specify a proper function b = b(a) of a and +20 + +(a) b ≥ a + µ +(b) b < a + µ +Figure 1: Intuition for the Optimal Value z∗ +consider the limit of the relative error as a → ∞, which is defined as +lim +a→∞ +z∗(a, b(a)) − ¯F(b(a)) +¯F(b(a)) +. +(10) +Heuristically, the larger the value of (10), the more conservative the RO approach is. We +will present the selection of b(a) in later subsections. +Now we consider estimating tail quantiles. Similarly, we can apply the RO approach +to get a worst-case estimation. More specifically, given the information of F(x) for x ≤ a, +suppose that our goal is to estimate q = F −1(p) where p ≥ 1−F(a) = 1−β. In order to get +a worst-case estimation, we maximize q among all the potential convex tail extrapolations. +That is, the worst-case estimation for q is obtained by solving +max +f +q +s.t. +� q +−∞ +f(x)dx = p, +� ∞ +a +f(x)dx = β, +f(a) = f(a+) = η, +f ′ ++(a) ≥ −ν, +f convex for x ≥ a, +f(x) ≥ 0 for x ≥ a. +(11) +21 + +We define +q∗ := inf{b ≥ a : z∗(a, b) = 1 − p}. +(12) +The curve in Figure 2 reflects the shape of z∗ against b. For p1 such that β − η2/(2ν) ≤ +1−p1 ≤ β, we may find a corresponding point q∗ +1 such that z∗(a, q∗ +1) = 1−p1. In particular, +if 1 − p1 = β − η2/(2ν), then by the definition in (12), we have that q∗ +1 = a + µ. However, +for p2 such that 1 − p2 < β − η2/(2ν), the corresponding q∗ +2 is defined as ∞. The following +theorem justifies this intuitive definition of q∗. +Figure 2: The Shape of z∗ against b and the Definition of q∗ +Theorem 3.2. q∗ defined in (12) is the optimal value of (11), and is expressed by +q∗ = +� +� +� +� +� +� +� +� +� +a + µ − +� +µ2 − σ + 2(1−p) +ν +if 1 − β ≤ p ≤ 1 − β + η2 +2ν; +∞ +if p > 1 − β + η2 +2ν. +(13) +Similar to estimation of tail probabilities, we manage to find a proper function p = p(a) +of a and compute the limit of the relative error as a → ∞, which is defined as +lim +a→∞ +q∗(p(a)) − q(p(a)) +q(p(a)) +. +(14) +Recall that q = q(p(a)) := F −1(p(a)) is the true p-quantile. Similarly, the larger the value +of (14), the more conservative the RO approach in estimating tail quantiles is. +22 + +3.2 +Conservativeness in Estimating Tail Probabilities +We consider two cases as follows. +Case 1: ξ > 0. Suppose that F and f, the true distribution function and density func- +tion of X, satisfy Assumptions 3.1 and 3.2 with ξ > 0. Since ¯F ∈ RV−1/ξ, by the Karamata +representation theorem (Embrechts et al., 1997), ¯F has the following representation: +¯F(x) = c(x) exp +� +− +� x +z +1 +u(t)dt +� +, z < x < ∞ +(15) +where c(x) → c > 0, u(x)/x → ξ as x → ∞. Moreover, it is known that when the threshold +a is sufficiently large, +P(X > x|X > a) ≈ ¯Gξ;a,u(a)(x) = +� +1 + ξx − a +u(a) +�− 1 +ξ +, x ≥ a, +which is exactly the mathematical foundation for the GPD method. +Heuristically, if P(X > x|X > a) is exactly equal to ¯Gξ;a,u(a)(x) for any x ≥ a, then we +get that +¯F(x) = P(X > a) ¯Gξ;a,u(a)(x) = β +� +1 + ξx − a +u(a) +�− 1 +ξ +. +Thus, +f(x) = − d +dxP(X > x) = +β +u(a) +� +1 + ξx − a +u(a) +�− 1 +ξ −1 +, +−f ′(x) = − d +dxf(x) = (ξ + 1)β +u2(a) +� +1 + ξx − a +u(a) +�− 1 +ξ −2 +. +If we substitute x with a, then we get that +η = +β +u(a), ν = (ξ + 1)β +u2(a) . +Therefore, it seems reasonable to use β/u(a) and (ξ + 1)β/u2(a) to approximate η and ν +respectively. Indeed, this is true as a → ∞, which is justified by the following proposition: +Proposition 3.3. Suppose that distribution function F and the corresponding density func- +tion f satisfy Assumptions 3.1 and 3.2 with ξ > 0, which implies that ¯F has the represen- +23 + +tation (15). The following statements are true: +lim +x→∞ +f(x)u(x) +¯F(x) += 1; lim +x→∞ − f ′(x)u2(x) +(ξ + 1) ¯F(x) = 1. +(16) +Remark. Note that this theorem implies that as a → ∞, +η ∼ +β +u(a), ν ∼ (ξ + 1)β +u2(a) , µ ∼ u(a) +ξ + 1, σ ∼ 2u2(a) +ξ + 1 . +(17) +Throughout this section, we use g1(a) ∼ g2(a) to denote lima→∞ g1(a)/g2(a) = 1. +For simplicity, we first consider b = a + xu(a) where x ≥ 0 is a fixed number. In this +case, it is known that ¯F(b)/ ¯F(a) → (1+ξx)−1/ξ as a → ∞ (Embrechts et al., 1997). Using +the conclusions in Proposition 3.3 above, we can get the following theorem: +Theorem 3.3. Suppose that distribution function F and the corresponding density function +f satisfy Assumptions 3.1 and 3.2 with ξ > 0, which implies that ¯F has the representation +(15). b = b(a) is chosen as b = a + xu(a) where x ≥ 0 is a fixed number. Then we have +that +lim +a→∞ +z∗(a, b) +¯F(b) += +� +� +� +� +� +� +� +� +� +� +1 − +1 +2(ξ+1) +� +(1 + ξx) +1 +ξ +if x ≥ +1 +ξ+1; +� +1 − x + ξ+1 +2 x2� +(1 + ξx) +1 +ξ +if x < +1 +ξ+1. +(18) +In fact, we may generalize the results in Theorem 3.3 to the case that b = a + x(a)u(a) +where lima→∞ x(a) = x0 ≥ 0. +By Proposition 0.5 in Resnick (1987), we get that the +convergence limx→∞ +¯F(tx) +¯F(x) = t− 1 +ξ holds locally uniformly on (0, ∞). Since the limit t− 1 +ξ is +continuous in t, continuous convergence holds. We have that +lim +a→∞ +a + x(a)u(a) +a += 1 + ξx0 > 0, +so +lim +a→∞ +¯F(a + x(a)u(a)) +¯F(a) += (1 + ξx0)− 1 +ξ . +Moreover, we can also follow the discussions in the above proof to get the limit of z∗/ ¯F(a), +and hence the limit of z∗/ ¯F(b). +For example, if we choose b = 2a, then we can set +24 + +x(a) = a/u(a) → 1/ξ. Since 1/ξ > 1/(ξ + 1), we get that +lim +a→∞ +z∗ +¯F(b) = +� +1 − +1 +2(ξ + 1) +� +2 +1 +ξ . +(19) +We note that (19) is decreasing in ξ for ξ > 0, and it converges to 1 as ξ → ∞. This shows +that the DRO approach is less conservative in estimating tail probabilities for heavier-tailed +distributions. +Case 2: ξ = 0 and xF = ∞. Now we suppose that F and f satisfy Assumption 3.1, +Assumption 3.2 with ξ = 0, and additionally Assumption 3.3. Substituting ξ with 0 in +(17), we naturally guess that as a → ∞, +η ∼ +β +u(a), ν ∼ +β +u2(a), µ ∼ u(a), σ ∼ 2u2(a). +(20) +This guess is justified by the proposition below. +Proposition 3.4. Suppose that distribution function F and the corresponding density func- +tion f satisfy Assumption 3.1, Assumption 3.2 with ξ = 0 and Assumption 3.3. We have +that ¯F is a von Mises function with the following representation: +¯F(x) = c exp +� +− +� x +z +1 +u(t)dt +� +, z < x < ∞ +(21) +where c is some positive constant and u(x) = ¯F(x)/f(x) is positive and absolutely contin- +uous with limx→∞ u′(x) = 0. In addition, the following statements hold: +lim +x→∞ +f(x)u(x) +¯F(x) += 1; +(22) +lim +x→∞ −f ′(x)u2(x) +¯F(x) += 1. +(23) +Similar to Case 1, we may choose b = a + xu(a) where x ≥ 0 is a fixed number, and +then we have the following results: +Theorem 3.4. Suppose that distribution function F and the corresponding density function +f satisfy Assumption 3.1, Assumption 3.2 with ξ = 0 and additionally, Assumption 3.3. +25 + +b = b(a) is chosen as b = a + xu(a) where x ≥ 0 is a fixed number. Then we have that +lim +u→∞ +z∗(a, b) +¯F(b) += +� +� +� +� +� +� +� +� +� +1 +2ex +if x ≥ 1; +� +1 − x + 1 +2x2� +ex +if x < 1. +(24) +Indeed, lima→∞ u(a)/a = 0, so no matter how large is x, we always have that (b − +a)/a = xu(a)/a → 0 as a → ∞. Thus if we choose b = 2a instead, then for any x ≥ 0, +lima→∞ ¯F(b)/ ¯F(a) ≤ lima→∞ ¯F(a + xu(a))/ ¯F(a) = e−x, and hence ¯F(b)/ ¯F(a) → 0 as +a → ∞. We also know that in this case z∗(a, b)/ ¯F(a) → 1/2. Therefore, in the Gumbel +case, z∗(a, 2a)/ ¯F(2a) → ∞ as a → ∞. Compared with (19), we conclude that in estimating +the tail probabilities, the DRO approach is more conservative in the light-tail case than in +the heavy-tail case. +3.3 +Conservativeness in Estimating Tail Quantiles +We again consider two cases as follows. +Case 1: ξ > 0. Suppose that F and f satisfy Assumption 3.1 and Assumption 3.2 with +ξ > 0. For simplicity, we choose p = p(a) such that 1 − p = xβ where 1 − +1 +2(ξ+1) < x ≤ 1 is +a fixed number. Recall that q∗ = ∞ if p > 1 − β + η2/(2ν). Using (17), we get that +lim +a→∞ +� +β − η2 +2ν +� +/β = 1 − +1 +2(ξ + 1). +Thus the requirement x > 1− +1 +2(ξ+1) guarantees that 1−p = xβ > β−η2/(2ν) for sufficiently +large a, and hence the RO approach can give a non-trivial estimate. +Theorem 3.5. Suppose that distribution function F and the corresponding density function +f satisfy Assumption 3.1 and Assumption 3.2 with ξ > 0. p is chosen as 1 − p = xβ where +1 − +1 +2(ξ+1) < x ≤ 1 is a fixed number. Then we have that +lim +a→∞ +q∗ +q = xξ +� +ξ +ξ + 1 +� +1 − +� +1 − 2(1 − x)(ξ + 1) +� ++ 1 +� +. +(25) +26 + +We note that as ξ grows, the feasible interval for x, i.e. (1− +1 +2(ξ+1), 1], becomes narrower. +We also note that (25) has an upper bound which only depends on ξ. Roughly speaking, +for any ξ > 0 and 1− +1 +2(ξ+1) < x ≤ 1, the value of (25) is always bounded by +ξ +ξ+1 +1, which +is increasing with ξ. Thus, for heavier-tailed distributions, estimating the tail quantiles is +more conservative. +Case 2: ξ = 0 and xF = ∞. Now we suppose that F and f satisfy Assumption 3.1, +Assumption 3.2 with ξ = 0, and additionally, Assumption 3.3. Similarly, we choose p such +that 1 − p = xβ where 1/2 < x ≤ 1 is a fixed number, which guarantees that q∗ < ∞ for +sufficiently large a. In this case, we have the following theorem: +Theorem 3.6. Suppose that distribution function F and the corresponding density function +f satisfy Assumption 3.1, Assumption 3.2 with ξ = 0 and additionally, Assumption 3.3. p +is chosen as 1 − p = xβ where 1/2 < x ≤ 1 is a fixed number. Then we have that +lim +a→∞ +q∗ +q = 1. +(26) +Compared to Theorem 3.5, in the light-tail case not only the feasible interval for x is +wider, but also the limit of the relative error is smaller, and thus it is less conservative to +estimate tail quantiles. +3.4 +Examples +For Cases 1 and 2, we respectively consider the Pareto distribution and the standard normal +distribution as examples, which verify our main conclusions about the relationship between +the conservativeness of the DRO approach and the heaviness of the tail. +Example 3.1 (Pareto distribution). Suppose that X has a Pareto distribution with scale +parameter xm > 0 and shape parameter α > 0. +Then the tail distribution function is +¯F(x) = (xm/x)α for x ∈ [xm, ∞). It is known that F ∈ MDA(Hξ) where ξ = α−1 > 0. The +27 + +larger the α is, the smaller ξ is, and the lighter is the tail. In estimating tail probabilities, +we choose b = ka where k ≥ 1. Then by the discussion following Theorem 3.3, we get that +the limit of relative error is +lim +a→∞ +z∗(a, b(a)) − ¯F(b(a)) +¯F(b(a)) += +� +� +� +� +� +� +� +� +� +� +1 − +1 +2(ξ+1) +� +k +1 +ξ − 1 +if k ≥ 1 + +ξ +ξ+1; +� +1 − k−1 +ξ ++ (ξ+1)(k−1)2 +2ξ2 +� +k +1 +ξ − 1 +if k < 1 + +ξ +ξ+1. +In estimating tail quantiles, we choose 1 − p = xβ where 1 − +1 +2(ξ+1) < x ≤ 1. Then by +Theorem 3.5, we get that the limit of relative error is +lim +a→∞ +q∗(p(a)) − q(p(a)) +q(p(a)) += xξ +� +ξ +ξ + 1 +� +1 − +� +1 − 2(1 − x)(ξ + 1) +� ++ 1 +� +− 1. +Figure 3 (a) shows how the limit of relative error changes with k and α in estimating +probabilities while (b) shows the change with x and α in estimating quantiles. It can be +seen that in estimating probabilities, heavier tail gives less conservativeness; however, in +estimating quantiles, for the heavier tail, the limit of relative error is larger and meanwhile +the range of x that gives non-trivial estimate q∗ is smaller, so the DRO approach is more +conservative. +(a) Estimating Tail Probabilities +(b) Estimating Tail Quantiles +Figure 3: Limit of Relative Errors for the Pareto Distribution +Example 3.2 (Standard normal distribution). Suppose that X has a standard normal +distribution. It is known that the distribution function F ∈ MDA(Hξ) where ξ = 0. In +28 + +15 +Q=1 +α=2 +limit of relative error +α=3 +10 +5 +0 +1 +1.5 +2 +2.5 +3 +k0.12 +α=1 +0.1 +α=2 +α=3 +error +0.08 + of relative +0.06 +limit +0.04 +0.02 +0 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +xparticular, F is a von Mises function with auxiliary function u(x) = ¯F(x)/f(x) ∼ 1/x, x → +∞. In estimating tail probabilities, if b = a + xu(a), we always have b − a → 0 as a → ∞. +Nevertheless, the value in (24) still grows exponentially in x. Therefore, the DRO approach +is very conservative in estimating probabilities. In estimating tail quantiles, we may choose +1 − p = xβ for 1/2 < x ≤ 1. This range is larger than that for any ξ > 0, and yet the +limit of relative error is always equal to 0, which means that the DRO approach is less +conservative than in the Fr´echet case. +4 +Optimization Reformulations and Solution Tractabil- +ity +In this section, we focus mainly on ψ(P) as an expectation, i.e., EP[h(X)] for some function +h. When the target ψ(P) is a quantile, the analysis can be reduced to the expectation case +because ψ(P) = min{q : P(X ≤ q) ≥ p} can be written as ψ(P) = min{q : E[h(X)] ≥ +p − P(X ≤ a)} where h(X) = I(a ≤ X ≤ q), and the non-tail part P(X ≤ a) is supposedly +handleable by standard statistical tools such as via the empirical distribution. Therefore, +in finding the quantile, one could consider line search methods such as bisection on q to +obtain the minimum q such that the following holds: +max +P∈F(P(P,g,S)) E[I(a ≤ X ≤ q)] ≥ p − P(X ≤ a). +There exists some challenges in solving (6) as it is an infinite-dimensional optimization +problem, and the geometric shape constraint imposes extra complication. In the following +subsections, we show how we leverage techniques in the optimization literature to reduce +(6) into a moment problem which can then be dualized into standard solvable program +classes. +29 + +4.1 +Transformation to Moment Problems +Given ψ(P) = EP[h(X)], moment constraints configured by g and S, and shape information +P as either P1 +a,η or P2 +a,η,¯η,ν, (6) can be written as +P(h, P, g, S) : +max +P +EP[h(X)] subject to P ∈ P, EP[g(X)] ∈ S +(27) +where we have now highlighted the role of h in the notation P(h, P, g, S). Our first step +in handling (27) is to convert it to an equivalent moment problem: +Ma(H, G, S′) : max +X∼Q EQ[H(X)] subject to EQ[G(X)] ∈ S′, Q ∈ P[a, ∞) +(28) +where the set S′ is derived from S, P[a, ∞) denotes the class of probability distributions +with support [a, ∞), and function H and vector function G are derived from h and g +respectively. More precisely, we have the following result. +Theorem 4.1. Suppose h is bounded and each gi is bounded from below with support x ≥ a. +1. P(h, P1 +a,η, g, S) is equivalent to Ma +� +η ˜H, η ˜G, S +� +where ˜H(x) = +� x +a h(u)du and ˜Gi(x) = +� x +a gi(u)du for each component i in the vector function ˜G. A bijective transformation +between a feasible solution P of P(h, P1 +a,η, g, S) and Q of Ma(η ˜H, η ˜G, S) is given by +P ′ ++(x) = η(1 − Q(x)) (viewing P and Q as distribution functions). +2. P(h, P2 +a,η,¯η,ν, g, S) is equivalent to Ma +� +νH, +� +ν(x − a), νG⊤�⊤, SR(η, ¯η) × S +� +where +H(x) = +� x +a +� u +a h(v)dvdu and Gi(x) = +� x +a +� u +a gi(v)dvdu for each component i in the vec- +tor function G. A bijective transformation between a feasible solution P in P(h, P2 +a,η,¯η,ν, g, S) +and Q in Ma +� +νH, +� +ν(x−a), νG⊤�⊤, SR(η, ¯η)×S +� +is given by −P (2) ++ (x) = ν(1−Q(x)), +where P (2) ++ +denotes the second-order right derivative of P (viewing P and Q as dis- +tribution functions). +To illustrate Theorem 4.1, consider for example a tail interval probability as the objec- +30 + +tive, in which h(x) = I(L ≤ x ≤ R) for some given number L, R. We have +˜H(x) = (x − L)I(L ≤ x ≤ R) + (R − L)I(x ≥ R); +H(x) = 1 +2(x − L)2I(L ≤ x ≤ R) + (R − L)(x − R + L +2 +)I(x ≥ R). +If gi(x) = xi−1I(x ≥ a), i ≥ 1, we have +˜Gi(x) = (xi − ai)/i, Gi(x) = xi+1/(i2 + i) − aix/i + ai+1/(i + 1). +Theorem 4.1 shows that (6) is equivalent to a moment-constrained program, by iden- +tifying the decision variable (as a probability distribution) via a one-to-one map with a +probability distribution function with support on [a, ∞). We give two distinct methods to +prove Theorem 4.1. The first one is an integration-by-parts technique that involves replac- +ing the distribution function in the decision variable by its derivative. This approach is +built on Lam and Mottet (2017) that considers a more restrictive formulation. The second +method is Choquet’s theory, which in convex analysis implies the representation of any +point in a compact convex set by a mixture of its extreme points. In our context, the +stipulated class of probability distributions can be written as a mixture representation of +simpler distributions. This then allows one to rewrite the optimization problem in terms +of the mixture distribution as the decision variable, and subsequently remove the shape +constraint. Though the main idea of this method follows from some existing DRO works +(Popescu, 2005; Van Parys et al., 2016; Li et al., 2019), our theorem allows for general +moment set S and inequality constraints for the accompanying parameters in the shape +constraint P1 +a,η or P2 +a,η,¯η,ν, instead of singleton used in these works. +4.2 +Dualization to Semidefinite Programs +The moment problem (28) has a finite number of constraints but an infinite-dimensional de- +cision variable. In the following, we transform it into a dual program with finite-dimensional +31 + +decision variable but an infinite number of constraints, which we can further reduce to a +more tractable formulation. Since our set S′ in the moment problem (28) generally consists +of both ellipsoid and rectangular sets, we write our theorem in this generality as well. First, +we introduce the following assumption for guaranteeing strong duality: +Assumption 4.1 (Slater Condition). In Ma(H, G, S) where (Gj)d +j=1 are measurable func- +tions, there exists P ∈ P[a, ∞) such that EP[G(X)] lies in the interior of S. +We have the following duality result: +Theorem 4.2. The dual problem of Ma +� +H, (G⊤ +1 , G⊤ +2 )⊤, SE(µ, Σ, r)×SR(µ, ¯µ) +� +, with given +constant values µ, Σ, ¯µ, µ, a, function H and vector functions G1, G2, is +min +κ,λ1≥0,λ2≥0, +∥u∥2≤λ +κ + λ + r−1/2u⊤Σ−1/2µ + λ⊤ +1 ¯µ − λ⊤ +2 µ +(29a) +s.t. +− H(x) + r−1/2u⊤Σ−1/2G1(x) + (λ1 − λ2)⊤G2(x) + κ ≥ 0, ∀x ≥ a. (29b) +Here, κ, u, λ, λ1, λ2 are decision variables. The optimal value of (29) is at least that of +Ma +� +H, (G⊤ +1 , G⊤ +2 )⊤, SE(µ, Σ, r) × SR(µ, ¯µ) +� +and, under Assumption 4.1, they attain equal- +ity. +Theorem 4.2 shows the dual program of (28) when S is a combination of ellipsoid +and rectangle. The Slater condition in Assumption 4.1 that ensures strong duality can be +checked routinely case-by-case. Moreover, even if this condition does not hold, the ultimate +statistical guarantees provided by Theorems 2.1 and 2.2 are still valid since weak duality +allows us to obtain a more conservative bound. Theorem 4.2 follows immediately from the +duality theory of conic programs (e.g., Shapiro (2001)). +Putting Theorems 4.1 and 4.2 together, we can convert (27) with (S, P) being (SE, P1 +a,η), +(SR, P1 +a,η), (SE, P2 +a,η,¯η,ν) or (SR, P2 +a,η,¯η,ν) into the following dual program. +32 + +Corollary 4.1. Given any functions h, g satisfying the assumptions in Theorem 4.1, pa- +rameters η, ν, a, µ, µ, ¯µ, Σ ≻ 0 and ˜H(x) = +� x +a h(u)du, H(x) = +� x +a +� u +a h(v)dvdu, ˜Gi(x) = +� x +a gi(u)du, Gi(x) = +� x +a +� u +a gi(v)dvdu for each component i of g, we have +1. For problem P(h, g, SE(µ, Σ, r), P1 +a,η), the dual of the converted moment problem is +min +κ,∥u∥2≤λ κ + λ + r−1/2u⊤Σ− 1 +2µ s.t. − η ˜H(x) + ηu⊤Σ− 1 +2 ˜G(x) + κ ≥ 0, ∀x ≥ a. +2. For problem P(h, g, SR(µ, ¯µ), P1 +a,η), the dual of the converted moment problem is +min +κ,λ1≥0,λ2≥0 κ + λ⊤ +1 ¯µ − λ⊤ +2 µ, s.t. − η ˜H(x) + η(λ1 − λ2)⊤ ˜G(x) + κ ≥ 0, ∀x ≥ a. +3. For problem P(h, g, SE(µ, Σ, r), P2 +a,η,¯η,ν), the dual of the converted moment problem +is +min +κ,∥u∥2≤λ +δ1≥0,δ2≥0 +κ + λ + r−1/2u⊤Σ−1/2µ + δ1¯η − δ2η, s.t. − νH(x) + νu⊤Σ−1/2G(x) + ν(δ1 − δ2)(x − a) + κ ≥ 0, x ≥ a. +4. For problem P(h, g, SR(µ, ¯µ), P2 +a,η,¯η,ν), the dual of the converted moment problem is +min +κ,δ1≥0,δ2≥0, +λ1≥0,λ2≥0 +κ + λ⊤ +1 ¯µ − λ⊤ +2 µ + δ1¯η − δ2η, s.t. − νH(x) + ν(λ1 − λ2)⊤G(x) + ν(δ1 − δ2)(x − a) + κ ≥ 0, ∀x ≥ a. +In each case, the optimal value of the dual problem is at least that of the corresponding +P(h, g, S, P) and, under Assumption 4.1 applied to the corresponding moment problem, +they attain equality. +Program (29) and the specialized versions in Corollary 4.1 have infinite numbers of +constraints, with (29b) being a condition for any x ≥ a. +We can convert (29) into a +semidefinite program under suitable assumptions. +Theorem 4.3. If the constraint (29b) can be written as a series of polynomial inequalities, +i.e., +k +� +i=0 +y1ixi ≥ 0, ∀x ≥ a or +k +� +i=0 +y2ixi ≥ 0, ∀b ≤ x ≤ c, +(30) +then program (29) is equivalent to a mixed semidefinite and second-order cone program +33 + +(SDP-SOCP) +min +u,λ,κ,λ1≥0,λ2≥0, +V ⪰0,W ⪰0,∥Σ1/2u∥2≤λ +κ + λ + u⊤µ + λ⊤ +1 ¯µ − λ⊤ +2 µ +s.t. +� +i,j:i+j=2l−1 +vij = +� +i,j:i+j=2l−1 +wij = 0, +l = 1, · · · , k, +� +i,j:i+j=2l +vij = +k +� +r=l +�r +l +� +y1rar−l, +l = 0, · · · , k, +� +i,j:i+j=2l +wij = +l +� +m=0 +k+m−l +� +r=m +� r +m +��k − r +l − m +� +y2rbr−mcm, +l = 0, · · · , k +(31) +where V = [vij]i,j=0,··· ,k and W = [wij]i,j=0,··· ,k. +Theorem 4.3 shows that problem (28) can be formulated into a tractable SDP-SOCP +program when H(x) and G(x) belong to the polynomial function class. If h(x) and g(x) +are indicator functions or piecewise polynomial functions, then the polynomial class con- +dition is satisfied which guarantees the tractability. +Theorem 4.3 can be proved using +the semidefinite representation of moments (e.g., Lasserre (2009); Bertsimas and Popescu +(2005)). +In Section 5, we use Corollary 4.1 and Theorem 4.3 to produce our numerical results +in estimating extremal quantities. In Appendix B we also present some sensitivity analysis +tools for our DRO problems. +5 +Numerical Results +We illustrate the numerical performance of our DRO framework and compare with conven- +tional EVT tools. In our experiment, we consider synthetic data of size 500. The quantities +of interest are tail interval probabilities and quantiles, which we will specify later. In the +experiments, we generate samples from “true” distributions that range from light-tailed to +heavy-tailed, including 1) Gamma distribution with shape parameter 0.5 and scale parame- +34 + +ter 1, 2) log-normal distribution with mean parameter 0 and standard deviation parameter +1, 3) Pareto distribution with shape parameter 1.5 and scale parameter 1. We aim to obtain +a one-sided 95% confidence upper bound by using program (6). To approximate the cover- +age probability, we repeat each experiment 200 times, from which we would also output the +sample mean of the estimated confidence bounds. Calibration of parameters is conducted +by bootstrapping with a resample size 500, where for densities and their derivatives we use +the standard kernel estimator from the R package ks. +We conduct experiments with different 1) shape constraints, 2) moment constraints 3) +cutoff thresholds a, 4) objective functions, and also compare with POT. These experiments +aim to: 1) validate our methodology by demonstrating how it generates valid confidence +bounds under a wide range of settings, as supported by the statistical guarantees in Theo- +rems 2.1 and 2.2; 2) provide guidance for users in implementing our approach; 3) compare +our approach against existing methods like POT. Sections 5.1-5.3 will discuss results per- +tinent to the first two goals, while Section 5.4 will focus on the third goal above. +In the following tables, we use (D, χ2) to denote the setting of D-th order monotonicity +and ellipsoid moment constraint depicted in Section 2.1 where D ∈ {0, 1, 2}. Similarly, +(D, KS) denotes the setting of D-th order monotonicity and rectangular moment constraint. +5.1 +Selection of Shape and Moment Constraints +In Table 5.1, we consider the estimation of tail interval probabilities P(q0.99 ≤ X ≤ q0.995) +using the synthetic data set from three distributions, where q0.99 and q0.995 are theoreti- +cal 99th and 99.5th-percentile respectively. The threshold a is chosen as the 70th sample +percentile of this synthetic data set. In this and the following tables, the “Upper Bound” +and “Coverage Probability” columns respectively show the sample mean and the ratio of +35 + +coverage of the upper confidence bounds in the 200 repetitions, and the “Relative Ratio” +column is defined as the mean upper confidence bound divided by the true value. The +confidence intervals are given by sample mean ± 1.96 ∗ sample standard deviation/ +√ +200. +We note that in some cases the average coverage probability is 1, i.e., the confidence bound +covers the truth in all the 200 repetitions, in which case the sample standard deviation as +well as the confidence interval width are 0. +Table 5.1: Tail probability estimation under different constraint settings. The true value +is 0.005. +Data Source +Constraint Setting +Relative Ratio +Upper Bound +Coverage Probability +Gamma +(0, χ2) +14.46 (±0.57) +7.23 × 10−2 (±2.85 × 10−3) +1.000 (±0.000) +(1, χ2) +5.36 (±0.17) +2.68 × 10−2 (±8.26 × 10−4) +1.000 (±0.000) +(2, χ2) +3.05 (±0.08) +1.53 × 10−2 (±4.12 × 10−4) +1.000 (±0.000) +(0, KS) +14.17 (±0.12) +7.08 × 10−2 (±5.88 × 10−4) +1.000 (±0.000) +(1, KS) +6.81 (±0.07) +3.40 × 10−2 (±3.39 × 10−4) +1.000 (±0.000) +(2, KS) +4.12 (±0.04) +2.06 × 10−2 (±2.04 × 10−4) +1.000 (±0.000) +Lognorm +(0, χ2) +16.62 (±0.80) +8.31 × 10−2 (±3.99 × 10−3) +1.000 (±0.000) +(1, χ2) +6.52 (±0.26) +3.26 × 10−2 (±1.32 × 10−3) +1.000 (±0.000) +(2, χ2) +3.98 (±0.14) +1.99 × 10−2 (±7.11 × 10−4) +1.000 (±0.000) +(0, KS) +14.08 (±0.13) +7.04 × 10−2 (±6.49 × 10−4) +1.000 (±0.000) +(1, KS) +7.98 (±0.09) +3.99 × 10−2 (±4.33 × 10−4) +1.000 (±0.000) +(2, KS) +5.09 (±0.05) +2.55 × 10−2 (±2.72 × 10−4) +1.000 (±0.000) +Pareto +(0, χ2) +20.90 (±1.68) +1.05 × 10−1 (±8.39 × 10−3) +1.000 (±0.000) +(1, χ2) +9.85 (±0.76) +4.92 × 10−2 (±3.78 × 10−3) +0.995 (±0.010) +(2, χ2) +6.83 (±0.44) +3.41 × 10−2 (±2.21 × 10−3) +0.995 (±0.010) +(0, KS) +14.13 (±0.13) +7.07 × 10−2 (±6.30 × 10−4) +1.000 (±0.000) +(1, KS) +9.76 (±0.10) +4.88 × 10−2 (±5.15 × 10−4) +1.000 (±0.000) +(2, KS) +6.81 (±0.07) +3.40 × 10−2 (±3.65 × 10−4) +1.000 (±0.000) +In Table 5.2, we consider the estimation of quantile q0.99 using the synthetic data set +from three distributions. The threshold is still chosen as the 70th sample percentile. The +true value of the 99th-quantile of each distribution is shown in the table. We note that +(D, KS) cannot always obtain a valid quantile estimation due to the possible assignment +36 + +of probability mass at ∞, so we do not include these settings in this table. +Table 5.2: Quantile estimation under different constraint settings. +Data Source +Constraint Setting +Relative Ratio +Upper Bound +Coverage Probability +Gamma w. true quantile point 3.32. +(0, χ2) +2.13 (±0.04) +7.07 × 100 (±1.34 × 10−1) +1.000 (±0.000) +(1, χ2) +1.50 (±0.03) +4.96 × 100 (±9.66 × 10−2) +1.000 (±0.000) +(2, χ2) +1.39 (±0.03) +4.62 × 100 (±9.20 × 10−2) +0.995 (±0.010) +Lognorm w. true quantile point 10.24. +(0, χ2) +2.54 (±0.10) +2.60 × 101 (±1.05 × 100) +1.000 (±0.000) +(1, χ2) +1.80 (±0.07) +1.84 × 101 (±7.40 × 10−1) +1.000 (±0.000) +(2, χ2) +1.69 (±0.07) +1.73 × 101 (±6.97 × 10−1) +0.990 (±0.014) +Pareto w. true quantile point 21.54. +(0, χ2) +6.05 (±1.33) +1.30 × 102 (±2.87 × 101) +1.000 (±0.000) +(1, χ2) +3.92 (±0.71) +8.45 × 101 (±1.54 × 101) +0.990 (±0.014) +(2, χ2) +3.59 (±0.61) +7.73 × 101 (±1.31 × 101) +0.985 (±0.017) +Regarding the selection of shape constraints, for tail probability estimation problem in +Table 5.1, as D increases, we observe a decreasing upper bound and confidence interval +width. +For example, with Gamma data in Table 5.1, the upper bound decreases from +7.23 × 10−2 to 1.53 × 10−2 (the true value is 5.00 × 10−3) and the confidence interval width +decreases from 2.85 × 10−3 to 2.04 × 10−4 as the constraint setting changes from (0, χ2) to +(2, χ2), leading to a tighter result. Similarly, for quantile estimation problem in Table 5.2, +we observe that the upper bound and confidence interval width decrease when the shape +constraint becomes stronger. Again, with Gamma data, the upper bound decreases from +7.07 to 4.62 (the true value is 3.32) and the confidence interval width decreases from 0.134 +to 0.092. +Indeed, by assuming shape property, we restrict the feasible distribution to a smaller +set compared to that without shape assumptions (D = 0). Moreover, convexity (D = 2) +implies monotonicity (D = 1) so the former set is a subset of the latter. Our results also +show that the extra errors in the additional estimation tasks needed in calibrating the +parameters under the stronger shape conditions do not seem to outweigh the benefits of +37 + +imposing the stronger constraints. In practice, one could visualize the distribution around +the threshold a to evaluate the plausibility of the shape assumption. +Now we compare the moment constraints for tail probability estimation. From Table 5.1, +we can see that the difference between the two moment constraints is not as substantial as +the one among the three shape constraints. Without any shape constraint (i.e., D = 0), the +rectangular constraint has a smaller relative ratio than the ellipsoidal one for all the three +distributions. In the presence of shape constraint (i.e., D = 1, 2), the ellipsoidal constraint +is less conservative than the rectangular one for Gamma and log-normal distributions, +and only slightly more conservative for Pareto distribution. In fact, for the Pareto data +and D = 1, 2, we cannot reject the null hypothesis that (D, χ2) and (D, KS) are equally +conservative using Welch’s t-test with the significance level 0.05. Overall, if one decides +to choose D = 0 after observing the data, then the rectangular constraint seems a better +choice for the moment constraint. Otherwise, the ellipsoidal constraint should have a better +or comparable performance. +5.2 +Selection of Threshold +In this section, we compare different selections of the threshold. In Table 5.3, we consider +the estimation of tail interval probabilities P(q0.99 ≤ X ≤ q0.995) under constraint setting +(2, χ2). For each data distribution, we test four single cutoff thresholds: 60th, 70th, 80th, +90th sample percentiles. We also test using all four of them as multiple thresholds (see +Supplement A for details of using multiple thresholds). +In Table 5.4, we consider the +estimation of quantile q0.99 also under constraint setting (2, χ2) with different choices of +cutoff thresholds. +We observe that as the cutoff threshold increases, the result tends to be less conservative +38 + +Table 5.3: Tail probability estimation under different cutoff threshold(s). The true value +is 0.005. +Data Source +Constraint Setting +Relative Ratio +Upper Bound +Coverage Probability +Gamma +60th +3.130 (±0.088) +1.57 × 10−2 (±4.39 × 10−4) +1.000 (±0.000) +70th +3.053 (±0.082) +1.53 × 10−2 (±4.12 × 10−4) +1.000 (±0.000) +80th +2.948 (±0.077) +1.47 × 10−2 (±3.85 × 10−4) +0.995 (±0.010) +90th +2.823 (±0.077) +1.41 × 10−2 (±3.85 × 10−4) +0.995 (±0.010) +(60th, 70th, 80th, 90th) +2.946 (±0.079) +1.47 × 10−2 (±3.93 × 10−4) +0.995 (±0.010) +Lognorm +60th +4.113 (±0.147) +2.06 × 10−2 (±7.37 × 10−4) +1.000 (±0.000) +70th +3.984 (±0.142) +1.99 × 10−2 (±7.11 × 10−4) +1.000 (±0.000) +80th +3.831 (±0.140) +1.92 × 10−2 (±6.99 × 10−4) +1.000 (±0.000) +90th +3.629 (±0.129) +1.81 × 10−2 (±6.45 × 10−4) +0.995 (±0.010) +(60th, 70th, 80th, 90th) +3.792 (±0.132) +1.90 × 10−2 (±6.61 × 10−4) +1.000 (±0.000) +Pareto +60th +7.093 (±0.500) +3.55 × 10−2 (±2.50 × 10−3) +0.995 (±0.010) +70th +6.825 (±0.443) +3.41 × 10−2 (±2.21 × 10−3) +0.995 (±0.010) +80th +6.428 (±0.367) +3.21 × 10−2 (±1.84 × 10−3) +0.995 (±0.010) +90th +5.559 (±0.233) +2.78 × 10−2 (±1.17 × 10−3) +0.995 (±0.010) +(60th, 70th, 80th, 90th) +5.821 (±0.240) +2.91 × 10−2 (±1.20 × 10−3) +0.995 (±0.010) +for both tail probability estimation and quantile estimation. For example, for Gamma dis- +tribution, as the threshold increases from 60th sample percentile to 90th sample percentile, +the relative ratio decreases from 3.130 to 2.823 for tail probability estimation, and from +1.426 to 1.331 for quantile estimation. This phenomenon is reasonable as more information +is leveraged with a larger threshold. The performance of multiple thresholds lies in the +middle of the ones of single thresholds. Ideally, we should choose a relatively large thresh- +old given that the parameters could be calibrated well. However, in practice, it is usually +hard to evaluate which threshold satisfies this condition. Thus, using multiple thresholds +is also a reasonable choice as it is less sensitive to the selection. +39 + +Table 5.4: Quantile estimation under different cutoff threshold(s). +Data Source +Constraint Setting +Relative Ratio +Upper Bound +Coverage Probability +Gamma w. true quantile point 3.32 +60th +1.426 (±0.027) +4.73 × 100 (±8.95 × 10−2) +1.000 (±0.000) +70th +1.392 (±0.028) +4.62 × 100 (±9.20 × 10−2) +0.995 (±0.010) +80th +1.359 (±0.029) +4.51 × 100 (±9.47 × 10−2) +0.990 (±0.014) +90th +1.331 (±0.029) +4.41 × 100 (±9.76 × 10−2) +0.980 (±0.019) +(60th, 70th, 80th, 90th) +1.354 (±0.031) +4.49 × 100 (±1.02 × 10−1) +0.980 (±0.019) +Lognorm w. true quantile point 10.24 +60th +1.702 (±0.067) +1.74 × 101 (±6.89 × 10−1) +0.995 (±0.010) +70th +1.686 (±0.068) +1.73 × 101 (±6.97 × 10−1) +0.990 (±0.014) +80th +1.670 (±0.069) +1.71 × 101 (±7.04 × 10−1) +0.985 (±0.017) +90th +1.650 (±0.068) +1.69 × 101 (±6.97 × 10−1) +0.970 (±0.024) +(60th, 70th, 80th, 90th) +1.690 (±0.071) +1.73 × 101 (±7.30 × 10−1) +0.980 (±0.019) +Pareto w. true quantile point 21.54 +60th +3.637 (±0.648) +7.83 × 101 (±1.40 × 101) +0.990 (±0.014) +70th +3.587 (±0.607) +7.73 × 101 (±1.31 × 101) +0.985 (±0.017) +80th +3.542 (±0.588) +7.63 × 101 (±1.27 × 101) +0.985 (±0.017) +90th +3.347 (±0.450) +7.21 × 101 (±9.69 × 100) +0.985 (±0.017) +(60th, 70th, 80th, 90th) +3.584 (±0.552) +7.72 × 101 (±1.19 × 101) +0.985 (±0.017) +5.3 +Performance on Different Objective Functions +In this section, we aim to understand how the performance of our approach would change +with the objective function. In Table 5.5, we show the tail probability estimation results +for different intervals under constraint settings (2, χ2) and (2, KS). More specifically, the +target probability is chosen as P(qLHS ≤ X ≤ qLHS+0.005) where LHS takes different values +ranging from 0.90 to 0.99. That is, we keep the true probability value as 0.005, and the +interval [qLHS, qLHS+0.005] moves to the farther part of the tail as LHS increases. The cutoff +threshold is chosen as the 70th sample percentile. +From the table, we see that the relative ratio tends to increase as the interval is on the +farther tail from the threshold, i.e., as LHS increases. For instance, for Gamma data under +(2, χ2) constraints, the relative ratio increases from 1.714 to 3.053 as the left endpoint LHS +increases from 0.90 to 0.99. The same trend is observed for all the data distributions and +40 + +Table 5.5: Tail probability estimation under different objective functions. The target prob- +ability is P(qLHS ≤ X ≤ qLHS+0.005). +(2, χ2) +(2, KS) +Data Source +LHS Quantitle +Relative Ratio +Upper Bound +Coverage Probability +Relative Ratio +Upper Bound +Coverage Probability +Gamma +0.900 +1.714 (±0.010) +8.57 × 10−3 (±4.94 × 10−5) +1.000 (±0.000) +1.766 (±0.014) +8.83 × 10−3 (±7.14 × 10−5) +1.000 (±0.000) +0.910 +1.782 (±0.010) +8.91 × 10−3 (±4.87 × 10−5) +1.000 (±0.000) +1.780 (±0.014) +8.90 × 10−3 (±7.12 × 10−5) +1.000 (±0.000) +0.920 +1.860 (±0.010) +9.30 × 10−3 (±4.92 × 10−5) +1.000 (±0.000) +1.806 (±0.015) +9.03 × 10−3 (±7.40 × 10−5) +1.000 (±0.000) +0.930 +1.950 (±0.010) +9.75 × 10−3 (±5.24 × 10−5) +1.000 (±0.000) +1.844 (±0.016) +9.22 × 10−3 (±7.88 × 10−5) +1.000 (±0.000) +0.940 +2.051 (±0.012) +1.03 × 10−2 (±6.14 × 10−5) +1.000 (±0.000) +1.899 (±0.017) +9.50 × 10−3 (±8.55 × 10−5) +1.000 (±0.000) +0.950 +2.163 (±0.016) +1.08 × 10−2 (±8.09 × 10−5) +1.000 (±0.000) +1.982 (±0.019) +9.91 × 10−3 (±9.35 × 10−5) +1.000 (±0.000) +0.960 +2.278 (±0.023) +1.14 × 10−2 (±1.16 × 10−4) +1.000 (±0.000) +2.111 (±0.021) +1.06 × 10−2 (±1.03 × 10−4) +1.000 (±0.000) +0.970 +2.390 (±0.033) +1.20 × 10−2 (±1.63 × 10−4) +1.000 (±0.000) +2.326 (±0.023) +1.16 × 10−2 (±1.15 × 10−4) +1.000 (±0.000) +0.980 +2.565 (±0.041) +1.28 × 10−2 (±2.04 × 10−4) +1.000 (±0.000) +2.755 (±0.028) +1.38 × 10−2 (±1.39 × 10−4) +1.000 (±0.000) +0.990 +3.053 (±0.082) +1.53 × 10−2 (±4.12 × 10−4) +1.000 (±0.000) +4.115 (±0.041) +2.06 × 10−2 (±2.04 × 10−4) +1.000 (±0.000) +Lognorm +0.900 +1.860 (±0.011) +9.30 × 10−3 (±5.52 × 10−5) +1.000 (±0.000) +1.951 (±0.016) +9.75 × 10−3 (±7.75 × 10−5) +1.000 (±0.000) +0.910 +1.959 (±0.011) +9.80 × 10−3 (±5.45 × 10−5) +1.000 (±0.000) +1.987 (±0.016) +9.93 × 10−3 (±7.91 × 10−5) +1.000 (±0.000) +0.920 +2.080 (±0.011) +1.04 × 10−2 (±5.49 × 10−5) +1.000 (±0.000) +2.038 (±0.017) +1.02 × 10−2 (±8.39 × 10−5) +1.000 (±0.000) +0.930 +2.225 (±0.011) +1.11 × 10−2 (±5.71 × 10−5) +1.000 (±0.000) +2.102 (±0.018) +1.05 × 10−2 (±9.04 × 10−5) +1.000 (±0.000) +0.940 +2.402 (±0.013) +1.20 × 10−2 (±6.50 × 10−5) +1.000 (±0.000) +2.185 (±0.020) +1.09 × 10−2 (±9.93 × 10−5) +1.000 (±0.000) +0.950 +2.613 (±0.018) +1.31 × 10−2 (±8.86 × 10−5) +1.000 (±0.000) +2.298 (±0.022) +1.15 × 10−2 (±1.10 × 10−4) +1.000 (±0.000) +0.960 +2.854 (±0.030) +1.43 × 10−2 (±1.48 × 10−4) +1.000 (±0.000) +2.471 (±0.025) +1.24 × 10−2 (±1.23 × 10−4) +1.000 (±0.000) +0.970 +3.109 (±0.051) +1.55 × 10−2 (±2.57 × 10−4) +1.000 (±0.000) +2.748 (±0.029) +1.37 × 10−2 (±1.43 × 10−4) +1.000 (±0.000) +0.980 +3.399 (±0.078) +1.70 × 10−2 (±3.89 × 10−4) +1.000 (±0.000) +3.305 (±0.036) +1.65 × 10−2 (±1.80 × 10−4) +1.000 (±0.000) +0.990 +3.984 (±0.142) +1.99 × 10−2 (±7.11 × 10−4) +1.000 (±0.000) +5.091 (±0.054) +2.55 × 10−2 (±2.72 × 10−4) +1.000 (±0.000) +Pareto +0.900 +2.204 (±0.014) +1.10 × 10−2 (±6.80 × 10−5) +1.000 (±0.000) +2.252 (±0.018) +1.13 × 10−2 (±9.23 × 10−5) +1.000 (±0.000) +0.910 +2.338 (±0.013) +1.17 × 10−2 (±6.42 × 10−5) +1.000 (±0.000) +2.306 (±0.019) +1.15 × 10−2 (±9.70 × 10−5) +1.000 (±0.000) +0.920 +2.511 (±0.013) +1.26 × 10−2 (±6.28 × 10−5) +1.000 (±0.000) +2.370 (±0.020) +1.19 × 10−2 (±1.02 × 10−4) +1.000 (±0.000) +0.930 +2.739 (±0.013) +1.37 × 10−2 (±6.39 × 10−5) +1.000 (±0.000) +2.456 (±0.022) +1.23 × 10−2 (±1.10 × 10−4) +1.000 (±0.000) +0.940 +3.042 (±0.014) +1.52 × 10−2 (±6.83 × 10−5) +1.000 (±0.000) +2.578 (±0.024) +1.29 × 10−2 (±1.19 × 10−4) +1.000 (±0.000) +0.950 +3.453 (±0.017) +1.73 × 10−2 (±8.53 × 10−5) +1.000 (±0.000) +2.739 (±0.026) +1.37 × 10−2 (±1.32 × 10−4) +1.000 (±0.000) +0.960 +4.013 (±0.032) +2.01 × 10−2 (±1.58 × 10−4) +1.000 (±0.000) +2.975 (±0.030) +1.49 × 10−2 (±1.50 × 10−4) +1.000 (±0.000) +0.970 +4.759 (±0.077) +2.38 × 10−2 (±3.85 × 10−4) +1.000 (±0.000) +3.375 (±0.034) +1.69 × 10−2 (±1.71 × 10−4) +1.000 (±0.000) +0.980 +5.691 (±0.186) +2.85 × 10−2 (±9.28 × 10−4) +1.000 (±0.000) +4.178 (±0.045) +2.09 × 10−2 (±2.23 × 10−4) +1.000 (±0.000) +0.990 +6.825 (±0.443) +3.41 × 10−2 (±2.21 × 10−3) +0.995 (±0.010) +6.808 (±0.073) +3.40 × 10−2 (±3.65 × 10−4) +1.000 (±0.000) +constraint settings. +Thus, when we infer the tail region from non-tail data, it is more +conservative if the target quantity is associated with farther tail. In this case, one could +try to increase the threshold if possible as discussed in Section 5.2, which could reduce the +conservativeness to certain degree. Otherwise, one could at least get a conservative but +safe estimation with this approach. +41 + +5.4 +Comparison with POT +Finally, we compare our approach with the conventional POT method, where a GPD is +fitted from the excess-loss data using maximum likelihood (e.g., Smith (1987)) and a 95% +confidence upper bound for the tail interval probability is then obtained from the delta +method. Table 5.6 shows the POT results, where we choose the threshold for fitting the +GPD according to the graphical approach based on the linearity of the mean excess function +(see Embrechts et al. (1997)). +Comparing Tables 5.5 and 5.6, we see that while our DRO method obtains looser bounds +than POT, it exhibits correct coverage. By contrast, POT undercovers (bold in Table 5.6) +in many cases. For instance, for an objective function P(q0.90 ≤ X ≤ q0.905) and Gamma +dataset, POT gives an upper confidence bound 4.64 × 10−3 which is even smaller than the +truth, while DRO gives 8.57 × 10−2 with configuration (2, χ2). On the other hand, POT +gives only 65% coverage while DRO gives 100% coverage. The subpar coverage of POT +suggests that the data size is too small to carry out proper estimation. +Overall, POT gives estimates closer to the true target quantity but its confidence bounds +can fall short of the prescribed coverage. Our recommendation is that a modeler whose +priority is about the order of magnitude would be better off choosing GPD, whereas a more +risk-averse modeler seeking a bound with correct confidence guarantee would be better off +choosing our DRO approach. +References +Atar, R., Chowdhary, K., and Dupuis, P. (2015). Robust bounds on risk-sensitive function- +als via r´enyi divergence. SIAM/ASA Journal on Uncertainty Quantification, 3(1):18–33. +42 + +Table 5.6: Tail probability estimation under different objective functions with the POT +method. The target probability is P(qLHS ≤ X ≤ qLHS+0.005). The coverage +probabilities under the nominal confidence level are bold in the table. +Data Source +LHS Quantitle +Relative Ratio +Upper Bound +Coverage Probability +Gamma +0.900 +0.928 (±0.060) +4.64 × 10−3 (±3.01 × 10−4) +0.650 (±0.066) +0.910 +1.096 (±0.046) +5.48 × 10−3 (±2.32 × 10−4) +0.780 (±0.057) +0.920 +1.159 (±0.040) +5.79 × 10−3 (±2.01 × 10−4) +0.850 (±0.049) +0.930 +1.177 (±0.035) +5.88 × 10−3 (±1.75 × 10−4) +0.880 (±0.045) +0.940 +1.170 (±0.032) +5.85 × 10−3 (±1.59 × 10−4) +0.905 (±0.041) +0.950 +1.164 (±0.030) +5.82 × 10−3 (±1.49 × 10−4) +0.895 (±0.042) +0.960 +1.169 (±0.030) +5.85 × 10−3 (±1.50 × 10−4) +0.865 (±0.047) +0.970 +1.210 (±0.032) +6.05 × 10−3 (±1.58 × 10−4) +0.900 (±0.042) +0.980 +1.346 (±0.037) +6.73 × 10−3 (±1.86 × 10−4) +0.950 (±0.030) +0.990 +1.705 (±0.056) +8.52 × 10−3 (±2.81 × 10−4) +0.970 (±0.024) +Lognorm +0.900 +1.002 (±0.051) +5.01 × 10−3 (±2.55 × 10−4) +0.730 (±0.062) +0.910 +1.110 (±0.037) +5.55 × 10−3 (±1.87 × 10−4) +0.830 (±0.052) +0.920 +1.185 (±0.023) +5.93 × 10−3 (±1.16 × 10−4) +0.910 (±0.040) +0.930 +1.200 (±0.020) +6.00 × 10−3 (±1.01 × 10−4) +0.935 (±0.034) +0.940 +1.219 (±0.019) +6.09 × 10−3 (±9.45 × 10−5) +0.965 (±0.025) +0.950 +1.241 (±0.020) +6.20 × 10−3 (±9.86 × 10−5) +0.955 (±0.029) +0.960 +1.272 (±0.022) +6.36 × 10−3 (±1.10 × 10−4) +0.950 (±0.030) +0.970 +1.331 (±0.024) +6.65 × 10−3 (±1.19 × 10−4) +0.975 (±0.022) +0.980 +1.461 (±0.029) +7.31 × 10−3 (±1.43 × 10−4) +0.990 (±0.014) +0.990 +1.731 (±0.045) +8.65 × 10−3 (±2.23 × 10−4) +0.985 (±0.017) +Pareto +0.900 +1.061 (±0.029) +5.30 × 10−3 (±1.44 × 10−4) +0.830 (±0.052) +0.910 +1.100 (±0.020) +5.50 × 10−3 (±1.00 × 10−4) +0.875 (±0.046) +0.920 +1.131 (±0.015) +5.66 × 10−3 (±7.45 × 10−5) +0.910 (±0.040) +0.930 +1.156 (±0.015) +5.78 × 10−3 (±7.37 × 10−5) +0.940 (±0.033) +0.940 +1.183 (±0.016) +5.92 × 10−3 (±7.81 × 10−5) +0.950 (±0.030) +0.950 +1.216 (±0.017) +6.08 × 10−3 (±8.45 × 10−5) +0.980 (±0.019) +0.960 +1.260 (±0.018) +6.30 × 10−3 (±9.19 × 10−5) +0.985 (±0.017) +0.970 +1.323 (±0.021) +6.62 × 10−3 (±1.04 × 10−4) +0.990 (±0.014) +0.980 +1.428 (±0.026) +7.14 × 10−3 (±1.30 × 10−4) +0.990 (±0.014) +0.990 +1.627 (±0.046) +8.13 × 10−3 (±2.28 × 10−4) +0.980 (±0.019) +43 + +Balkema, A. 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Given n sample points from d-dimensional random vectors X with +positive variance-covariance matrix Σ, according to weak convergence results, e.g., Corol- +lary 2.1 in Dik and de Gunst (1985), we have +n(E[X] − ˆµ)⊤ ˆΣ−1 +n (E[X] − ˆµ)⇒χ2 +d +(A.1) +where ˆµ and ˆΣn are the sample mean and sample covariance matrix respectively and χ2 +d is +a chi-squared distribution with d degrees of freedom. +2. Kolmogorov distribution. Given n i.i.d. realization ordered points {x1, · · · , xn} from +random variable X with continuous cumulative probability distribution F and the corre- +sponding empirical distribution Fn, the Kolmogorov-Smirnov statistic Dn +� +:= supx |Fn(x)− +F(x)| +� +converges to Kolmogorov distribution, e.g., Noether (1963), i.e., +√n max +i=1,··· ,n +� +max +�i − n +n ++ E[I(X ≥ xi)], n + 1 − i +n +− E[I(X ≥ xi)] +�� +⇒ sup +t∈[0,1] +|B(t) − tB(1)| +(A.2) +where B(t) is a standard Wiener process. +We discuss the calibration methods based on empirical observations to achieve the +statistical guarantee results in Theorem 2.1 and 2.2. To construct feasible regions that +satisfy coverage of true distribution Ptrue with high probability, one could calibrate the +parameters Γ, η, ν in a statistical perspective. Overall, S can be constructed based on the +aforementioned weak convergence results. The estimation of η and ν can be accessible +via kernel density estimation and boostrapping. +A Bonferroni correction is applied to +guarantee simultaneous confidence level of the estimation of those parameters. We illustrate +the procedure in both ellipsoidal and rectangular cases. +Consider a sample realization +1 + +{x1, x2, · · · , xn} from random variable X with cumulative distribution F. +Ellipsoidal Constraint +Denote ˆµ = 1 +n +�n +i=1 g(xi), ˆΣ = +1 +n−1 +�n +i=1 +� +g(xi) − ˆµ +�� +g(xi) − +ˆµ +�⊤ and assume the dimension of g is d. To calibrate the ellipsoidal region SE, we utilize +the χ2 weak convergence result depicted in (A.1). In particular, we need to determine z in +n(E[g(X)] − ˆµ)⊤ ˆΣ−1 +n (E[g(X)] − ˆµ) ≤ z, +to construct the required confidence region. +To that end, value z is chosen as the +� +1 − α +2 +�th or +� +1 − α +3 +�th quantile of chi-squared +distribution with d degrees of freedom under P1 +a,η or P2 +a,η,¯η,ν respectively. For the former +case, η is estimated as the 1 − α +2 percentile of the bootstrapped densities at a. For the +latter case, η and ¯η are chosen as the α +6 percentile and 1− α +6 percentile of the bootstrapped +densities at a respectively and −ˆν the α +3 percentile of the bootstrapped coefficients F (2) ++ (a). +In the case of multiple threshold levels, we seek to obtain a result which is the optimal +of all objective values ranging over different ai, i = 1, ..., m and cover the true value with +probability 1 − α. Under this scenario, value z is chosen as +� +1 − +α +m+1 +�th or +� +1 − +α +2m+1 +�th +quantile of chi-squared distribution with d degrees of freedom under P1 +ai,η or P2 +ai,η,¯η,ν +respectively. Alternatively, we may also choose z via bootstrapping, depicted in Algorithm +1. For each ai, i = 1, ..., m, ˆµ, ˆΣ are the empirical mean and covariance. η is chosen as the +1 − +α +m+1 percentile of the bootstrapped densities at ai for P1 +ai,η. For P2 +ai,η,¯η,ν, η and ¯η are +chosen as the +α +4m+2 percentile and 1 − +α +4m+2 percentile of the bootstrapped densities at ai +respectively and −ˆν the +α +2m+1 percentile of the bootstrapped coefficients F (2) ++ (ai). +2 + +Algorithm 1 Bootstrap procedure for computing z. +1: for b = 1, . . . , 500 do +2: +Randomly sample n data points with replacement from {x1, · · · , xn}. +3: +Compute ˆz after each sampling: +ˆz = +max +ai,i=1,...,m +� +n +�ˆˆµ − ˆµ +�T ˆΣ−1� +ˆˆµ − ˆµ +�� +where ˆˆµ is the empirical mean of {g(xb +i)}n +i=1 and {xb +1, · · · , xb +n} is the data set we obtain after this resam- +pling. +4: end for +5: Select the (1− +α +m+1)th or (1− +α +2m+1)th quantile of the ˆz’s obtained above under P1 +ai,η or P2 +ai,η,¯η,ν respectively. +Rectangular Constraint +To calibrate the rectangular region SR, we use the Kolmogorov- +Smirnov weak convergence result shown in (A.2). In particular, we need to determine z +in +√n max +i=1,··· ,n ∥F(xi) − �F(xi)∥∞ ≤ z, +(A.3) +where �F is the empirical distribution, to construct the confidence region. To this end, value +z is the +� +1 − α +2 +�th or +� +1 − α +3 +�th quantile of Kolmogorov distribution under P1 +a,η or P2 +a,η,ˆη,ν +respectively. For the former case, η is the 1 − α +2 percentile of the bootstrapped densities +F (1)(a). For the latter case, ¯η and η are the upper bound and lower bound of F (1)(a) with +joint probability 1 − α +3 and −ˆν is the lower bound of F (2) ++ (a|x ≥ a) with probability 1 − α +3 . +For the setting of choosing among a range of thresholds ai, i = 1, ..., m. Value z is +chosen as +� +1 − +α +m+1 +�th or +� +1 − +α +2m+1 +�th quantile of Kolmogorov distribution divided by √n +under P1 +ai,η or P2 +ai,η,ˆη,ν respectively. Alternatively, we may also choose z via bootstrapping, +depicted in Algorithm 2, where ∆ = 1 − +α +m+1 for P1 +ai,η and ∆ = 1 − +α +2m+1 for P2 +ai,η,¯η,ν. +For the first case, η is set as the 1 − +α +m+1 percentile of the bootstrapped densities F (1)(ai). +For the second case, ¯η and η are the upper bound and lower bound of F (1) ++ (ai) with joint +probability 1− +α +2m+1 and −ˆν is the lower bound of F (2) ++ (ai|x ≥ ai) with probability 1− +α +2m+1. +3 + +Algorithm 2 Bootstrap procedure for computing z. +1: for b = 1, . . . , 500 do +2: +Randomly sample n data points with replacement from {x1, · · · , xn}. +3: +Compute ˆz after each sampling: +ˆz = +max +ai,i=1,...,m +�√n +max +j=1,2,··· ,n +� +max +� j +n − +#{k : ai ≤ xk ≤ xb +j} +n +, +#{k : ai ≤ xk ≤ xb +j} +n +− j − 1 +n +��� +where {xb +1, · · · , xb +n} is the non-decreasing ordered dataset obtained after each sampling. +4: end for +5: Select the (∆)th quantile of the ˆz’s obtained above. +6: z is output as the selected (∆)th quantile in Step 5 divided by √n. +B +Sensitivity Analysis +On a high level, we give a perturbation analysis on the optimal value of the parameterized +problem Pµ in (A.4) with parameters µ. In particular, for a given perturbation direction +dr, we give a formula to express the perturbed optimal value of the problem as a linear +function of dr. This analysis shows how the DRO estimation changes with respect to the +calibration accuracy, and hence helps us understand how robust the estimation is to the +randomness in the data and the calibration procedure. +In program Pµ with given functions H(x) and Gj(x), ∀j ∈ {0} ∪ [d] for some positive +integer d where [d] := {1, . . . , d}, we assume (1). G0(x) = 1, µ0 = 1, restricting the non- +negative bounded measures to probability measures; (2). the parameters µj ∈ R, ∀j ∈ [d]. +Note that the dual of Pµ, i.e., Dµ, is a linear semi-infinite programming, in which there are +finite number of decision variables and infinite number of linear constraints. According to +Goberna et al. (1981), the relation a⊤x ≥ β with the associated vector (a⊤, β)⊤ is called +a linear consequence relation of the constraints system in some program P if every feasible +point in F(P) satisfies the relation. Program P is then Farkas-Minkowski (FM) if every +linear consequence relation of the constraints system of P is a linear consequence relation +of a finite subsystem. Lastly, for any program P, F(P), F∗(P) and v(P) denote the feasible +4 + +region, the optimal solution region and the optimal objective value of P respectively. +(Pµ) : +max +X∼P∈M +(Ω) EP [H(X)] +s.t. +EP [G(X)] = µ. +(A.4) +(Dµ) : min +y +y⊤µ +s.t. +y⊤G(x) ≥ H(x), ∀x ∈ Ω. +where M +(Ω) denotes the space of non-negative bounded measures on Ω. +Theorem A.1. Suppose Pµ is feasible and Dµ is a feasible FM system with µ ∈ int +� +M(Dµ) +� +, +then for any direction dr satisfying that the constraint system +� +EP[Gj(X)]+µjπ = drj, ∀j ∈ +{0} ∪ [d] +� +for (P, π) is non-empty, there exists ϵ > 0 such that ∀ρ : 0 ≤ ρ < ϵ, +v(Pµ+ρdr) = v(Pµ) + ρ min{dr⊤y|y ∈ F∗(Dµ)}. +(A.5) +The derivation mainly follows standard duality and sensitivity analysis for linear semi- +infinite linear optimization (Goberna and L´opez, 2000) seen in Theorem 2 of Goberna +et al. (2007). There are several sufficient conditions for Dµ to be an FM system. For +instance, if Ω is a compact set, function H and vector functions G are continuous such +that ∃y, y⊤G(x) > H(x), ∀x ∈ Ω, then Dµ is an FM system according to Lemma A.1. +Generally speaking, such a first-order expansion requires solving an auxiliary optimization +problem to obtain the difference v(Pµ+ρdr) − v(Pµ). However, if dr is a multiple of µ, +i.e., ρdr = cµ for some c ̸= 0, then the constraint system +� +EP[Gj(X)] + µjπ = drj, ∀j ∈ +{0} ∪ [d] +� +is non-empty for (P, π) and therefore (A.5) is simplified as +v(Pµ+dr) = v(P(1+c)µ) = v(Pµ) + c min{µ⊤y|y ∈ F∗(Dµ)} = (1 + c)v(Pµ). +5 + +C +Supplement: Proofs of Results +C.1 +Proof of Theorem 2.1 +Proof. For program P(P1 +a,η, g, S), if Ptrue ∈ F(P(P1 +a,η, g, S)), we will have ψ(Ptrue) ≤ +v(P(P1 +a,η, g, S)) where v(P) is the optimal value of program P. Therefore, +P(v(P(P1 +a,η, g, S)) ≥ ψ(Ptrue)) ≥ P(Ptrue ∈ F(P(P1 +a,η, g, S))) = 1 − α. +Similar arguments hold for P(P2 +a,η,¯η,ν, g, S). +C.2 +Proof of Theorem 2.2 +Proof. If Ptrue ∈ � +i=1,...,m F(P(Pi, gi, Si)), we have +ψ(Ptrue) ≤ v +� +P(Pi, gi, Si) +� +, i = 1, ..., m. +Hence +P +� +min +ai,i=1,...,m v +� +P(Pi, gi, Si) +� +≥ ψ(Ptrue) +� +≥P +� +Ptrue ∈ +� +i=1,...,m +F(P(Pi, gi, Si)) +� +=1 − α. +Similar arguments hold for Ptrue ∈ P2(mini ai). +C.3 +Proof of Proposition 3.1 +Proof. By definition, we know that Y > 0 with probability 1, and that +FY (x) = P (Y ≤ x) = P +� +X ≤ xF − x−1� += F +� +xF − x−1� +. +(A.6) +The fact that FY ∈ MDA(H−ξ) is proved in Embrechts et al. (1997). Now we prove the +remaining statements. By taking derivatives, we get that +fY (x) = f (xF − x−1) x−2, +(A.7) +f ′ +Y (x) = f ′ (xF − x−1) x−4 − 2f (xF − x−1) x−3. +(A.8) +6 + +By Assumption 3.1, f (xF − x−1) and f ′ (xF − x−1) exist for x > zY := 1/(xF − z) > 0. +Thus FY is twice differentiable on (zY , ∞). We also know that f (xF − x−1) > 0 for x > zY . +Then by (A.7), we get that fY > 0 on (zY , ∞). Moreover, fY is decreasing on (zY , ∞) since +f (xF − x−1) and x−2 are both positive and decreasing for x > zY . Now we only need to +prove that fY is also convex on (zY , ∞). Indeed, for any zY < x1 < x2 and 0 < λ < 1, we +have that +fY (λx1 + (1 − λ)x2) +=f +� +xF − +1 +λx1 + (1 − λ)x2 +� +1 +(λx1 + (1 − λ)x2)2 +≤f +� +xF − λ +x1 +− 1 − λ +x2 +� � λ +x2 +1 ++ 1 − λ +x2 +2 +� +≤ +� +λf +� +xF − x−1 +1 +� ++ (1 − λ)f +� +xF − x−1 +2 +�� � λ +x2 +1 ++ 1 − λ +x2 +2 +� +=λ2f +� +xF − x−1 +1 +� +x−2 +1 ++ (1 − λ)2f +� +xF − x−1 +2 +� +x−2 +2 ++ λ(1 − λ) +� +f +� +xF − x−1 +1 +� +x−2 +2 ++ f +� +xF − x−1 +2 +� +x−2 +1 +� +≤λ2f +� +xF − x−1 +1 +� +x−2 +1 ++ (1 − λ)2f +� +xF − x−1 +2 +� +x−2 +2 ++ λ(1 − λ) +� +f +� +xF − x−1 +1 +� +x−2 +1 ++ f +� +xF − x−1 +2 +� +x−2 +2 +� +=λfY (x1) + (1 − λ)fY (x2). +Therefore, FY also satisfies Assumption 3.1. +C.4 +Proof of Proposition 3.2 +Proof. Clearly, (A.6), (A.7) and (A.8) still hold. Then we may follow the proof of Proposi- +tion 3.1 to define zY and prove that on (zY , ∞), FY is twice differentiable and fY is positive, +decreasing and convex. The remainder of this proof inspires from Embrechts et al. (1997). +Since F satisfies Assumption 3.1 and 3.3, it is known that F is a von Mises function with +auxiliary function ¯F/f. More specifically, ¯F has the following representation: +¯F(x) = c exp +� +− +� x +z +f(t) +¯F(t)dt +� +, z < x < xF +where c is a positive constant. Then we have that +¯FY (x) = ¯F +� +xF − x−1� += c exp +� +− +� xF −x−1 +z +f(t) +¯F(t)dt +� +7 + += c exp +� +− +� x +1/(xF −z) +f (xF − s−1) s−2 +¯F (xF − s−1) ds +� += c exp +� +− +� x +zY +fY (s) +¯FY (s)ds +� +. +By definition, FY is also a von Mises function with auxiliary function ¯FY /fY , which implies +that FY ∈ MDA(Λ) and that (7) holds. +C.5 +Proof of Theorem 3.1 +Proof. Define h(x) = I(x > b). Correspondingly, +H(x) = +� x +0 +� t +0 +h(s + a)dsdt = (x − b + a)2 +2 +I(x > b − a). +Then Assumption 1 and Assumption 2 in Lam and Mottet (2017) hold. Indeed, h : R → +R+ is bounded and is nondecreasing in [a, b) and nonincreasing in (b, ∞). +Also, λ = +limx→∞ H(x)/x2 = 1 +2. Therefore, we can apply Theorem 4 in Lam and Mottet (2017). In +particular, z∗ = maxx∈[0,µ] W(x) where +W(x) = +� +� +� +� +� +� +� +� +� +ν +� +σ−µ2 +σ−2µx+x2H(x) + +(µ−x)2 +σ−2µx+x2H( σ−µx +µ−x ) +� +if x ∈ [0, µ); +ν(H(µ) + λ(σ − µ2)) +if x = µ. +If µ ≤ b − a, then it can be proved that arg max W(x) = µ and thus z∗ = W(µ) = +ν +2(σ − µ2). If µ > b − a, then it can be proved that arg max W(x) = [b − a, µ], and thus +z∗ = ν +2[σ − 2(b − a)µ + (b − a)2]. +C.6 +Proof of Theorem 3.2 +Proof. We denote the optimal value of (11) as qopt and then our goal is to show that +q∗ = qopt. First, we prove that qopt ≤ q∗. Indeed, for any feasible function ˜f, the p-quantile +q is the value that satisfies +� ∞ +q +˜f(x)dx = 1 − p. We see that ˜f is also feasible for (8), and +thus z∗(a, q) ≥ 1 − p. By the definition of q∗, we know that q ≤ q∗. Hence, qopt ≤ q∗. Now +we justify that qopt = q∗. +If β − η2/(2ν) < 1 − p ≤ β, then a ≤ q∗ < a + µ. Consider a feasible function ˜f that +8 + +decreases on [a, q∗] with derivative −ν. Then the p-quantile for ˜f is exactly q∗. +If 1 − p = β − η2/(2ν), then q∗ = a + µ. Consider a feasible function ˜f that decreases +on [a, a + µ − ε] with derivative −µ and then becomes extremely flat. Here, ε is a small +positive number. Then the p-quantile is between a + µ − ε and a + µ. As ε → 0, we get +that qopt = q∗. +If 1 − p < β − η2/(2ν), then q∗ = ∞. Still, we consider the same feasible function as in +the above case. Now the quantile q can be as large as we want, and hence qopt = ∞ = q∗. +In conclusion, q∗ defined in (12) is exactly the optimal value of (11). +Using (9), we can easily derive an explicit expression of q∗ since the non-constant part +of z∗ is actually a quadratic function. +C.7 +Proof of Proposition 3.3 +Proof. It is known that ¯F ∈ RV−1/ξ. We also have that F is absolutely continuous and +that (F)′ = −f is monotone. Then we can apply the Proposition 0.7 in Resnick (1987) +to get that f ∈ RV−1/ξ−1 and that limx→∞ − xf(x) +F(x) = − 1 +ξ. Also, since f is convex based on +our assumption, f is absolutely continuous and that f ′ is monotone. We can then apply +Proposition 0.7 again to get that limx→∞ +xf′(x) +f(x) = − 1 +ξ − 1. +In representation (15), we require that limx→∞ u(x)/x = ξ which gives +lim +x→∞ +f(x)u(x) +¯F(x) += lim +x→∞ +xf(x) +¯F(x) +u(x) +x += 1, +lim +x→∞ − f ′(x)u2(x) +(ξ + 1) ¯F(x) = lim +x→∞ − +1 +ξ + 1 +xf ′(x) +f(x) +xf(x) +¯F(x) +u2(x) +x2 += 1. +9 + +C.8 +Proof of Theorem 3.3 +Proof. If x > 1/(ξ + 1), then lima→∞(b − a)/µ = (ξ + 1)x > 1. Thus, for sufficiently large +a, we must have that b − a > µ. By Theorem 3.1, we get that z∗ = ν +2(σ − µ2). Hence, +lim +a→∞ +z∗(a, b) +¯F(a) += lim +a→∞ +z∗(a, b) +β += lim +a→∞ +� +1 − νµ2 +2β +� += 1 − +1 +2(ξ + 1). +If x < 1/(ξ + 1), then similarly, for sufficiently large a, we have that b − a < µ, and +thus z∗ = ν +2[σ − 2(b − a)µ + (b − a)2]. Hence, +lim +a→∞ +z∗(a, b) +¯F(a) += lim +a→∞ +z∗(a, b) +β += lim +a→∞ +� +1 − xu(a)νµ +β + x2u2(a) ν +2β +� += 1 − x + ξ + 1 +2 +x2. +Clearly, if x = 1/(ξ + 1), then the limit is the same as the case that x > 1/(ξ + 1). +Therefore, combining with lima→∞ ¯F(b)/ ¯F(a) = (1 + ξx)−1/ξ, we get (18). +C.9 +Proof of Proposition 3.4 +Proof. For the proof of representation (21) see Example 3.3.23 in Embrechts et al. (1997). +(22) follows directly from the expression of u(x). (23) follows from Assumption 3.3. +C.10 +Proof of Theorem 3.4 +Proof. Similar to the proof of Theorem 3.3, using the results in Proposition 3.4, we get +that +lim +a→∞ +z∗(a, b) +¯F(a) += lim +a→∞ +z∗(a, b) +β += +� +� +� +� +� +� +� +� +� +1 +2 +if x ≥ 1; +1 − x + 1 +2x2 +if x < 1. +Moreover, it is known that lima→∞ ¯F(b)/ ¯F(a) = e−x (Embrechts et al., 1997), and thus we +get (24). +10 + +C.11 +Proof of Theorem 3.5 +Proof. We have that +lim +a→∞ +q∗ +q = lim +a→∞ +(q∗ − a)/a + 1 +(q − a)/a + 1 . +First we deal with lima→∞(q∗ − a)/a. We have shown that for sufficiently large a, +q∗ = a + µ − +� +µ2 − σ + 2(1 − p) +ν +. +Then +lim +a→∞ +q∗ − a +a += lim +a→∞ +µ − +� +µ2 − σ + 2xβ +ν +µ +µ +u(a)/(ξ + 1) +u(a) +(ξ + 1)a = +ξ +ξ + 1 +� +1 − +� +1 − 2(1 − x)(ξ + 1) +� +. +Next we deal with lima→∞(q − a)/a. We define U(t) = F −1(1 − t−1), t > 0. Then we +can write q and a as +q = F −1(p) = U +� +1 +1 − p +� +, a = F −1(1 − β) = U +� 1 +β +� +. +It is known that (Embrechts et al., 1997) +lim +s→∞ +U(st) − U(s) +u(U(s)) += tξ − 1 +ξ +. +Note that +1 +1−p/ 1 +β = 1/x, so we set s = 1/β, t = 1/x and get +lim +a→∞ +q − a +u(a) = tξ − 1 +ξ += x−ξ − 1 +ξ +. +Hence lima→∞(q − a)/a = x−ξ − 1. By combining the two parts, we get (25). +C.12 +Proof of Theorem 3.6 +Proof. For sufficiently large a, we know that q∗ ≤ a+µ. On the other hand, since p ≥ 1−β, +we know that the true quantile q ≥ a. Thus +1 ≤ lim +a→∞ +q∗ +q ≤ lim +a→∞ +a + µ +a += 1 + lim +a→∞ +µ +u(a) +u(a) +a . +By Proposition 3.4, we know that µ/u(a) → 1. It is also known that u(a) → a = 0. +Therefore, we get (26). +11 + +C.13 +Proof of Theorem 4.1: Integration by parts method +Proof. For the first item of the theorem, without loss of generality, we assume that g = +� +I(x ≥ a), g(x) +�⊤ and S = {Γ} = {(β, Γ)⊤} for some scalars β, Γ where g(x) : [a, ∞) → R +is an integrable function over [a, ∞) and later we show how the result can be generalized +to any region S. We rewrite the program P(h, g, {Γ}, P1 +a,η) as +max +f +� ∞ +a +h(x)f(x)dx +(A.9a) +s.t. +� ∞ +a +f(x)dx = β, +(A.9b) +� ∞ +a +g(x)f(x)dx = Γ, +(A.9c) +f(a) ≤ η, +(A.9d) +f(x) exists, non-increasing and right-continuous for x ≥ a, +(A.9e) +f(x) ≥ 0 for x ≥ a. +(A.9f) +From assumptions that h(x) and g(x) only take nonzero values over [a, ∞), we can focus on +the integration starting from a. We consider f(x) as the right derivative of the cumulative +distribution function F. Since the set of discontinuous points of a monotone function is at +most countable which do not influence the integration values over [a, ∞), we assume f(x) +right-continuous for x ≥ a for all those discontinuous points. +Denote +˜H(x) = +� x +a +h(u)du, +˜G(x) = +� x +a +g(u)du. +As ˜H is continuous, ˜H(a) = 0 by definition, and f has bounded variation because of (A.9d),(A.9e) +and (A.9f), we have, using integration by parts, that (A.9a) is equal to +� ∞ +a +f(x)h(x)dx = +� ∞ +a +f(x)d ˜H(x) += f(x) ˜H(x)|∞ +a − +� ∞ +a +˜H(x)df(x) += − +� ∞ +a +˜H(x)df(x) +12 + +where the third equality follows from Lemma A.2 presented later with α = 0 and that +˜H(x) = O(x) as x → ∞ since h is bounded. +(A.9b) can be rewritten as +� ∞ +a +f(x)dx = +� ∞ +a +f(x)d(x − a) += f(x)(x − a)|∞ +a − +� ∞ +a +(x − a)df(x) += − +� ∞ +a +(x − a)df(x) +where the third equality follows from Lemma A.2 again with α = 0. +For (A.9c), as ˜G is continuous and ˜G(a) = 0 based on definition, we can write +� ∞ +a +f(x)g(x)dx = +� ∞ +a +f(x)d ˜G(x) += f(x) ˜G(x)|∞ +a − +� ∞ +a +˜G(x)df(x) += − +� ∞ +a +˜G(x)df(x) +where the third equality follows from Lemma A.3. +Finally, since f(x) → 0 as x → ∞ by Lemma A.2 with α = 0, we can write (A.9d) as +f(a) = − +� ∞ +a +df(x). +Therefore, (A.9) is equivalent to +max +f +− +� ∞ +a +˜H(x)df(x), +(A.10a) +s.t. +− +� ∞ +a +(x − a)df(x) = β, +(A.10b) +− +� ∞ +a +˜G(x)df(x) = Γ, +(A.10c) +− +� ∞ +a +df(x) ≤ η, +(A.10d) +f(x) exists, non-increasing and right-continuous for x ≥ a, +(A.10e) +f(x) ≥ 0 for x ≥ a, +(A.10f) +xf(x) → 0 as x → ∞, +(A.10g) +13 + +˜G(x)f(x) → 0 as x → ∞. +(A.10h) +The equivalence of (A.9) and (A.10) can be checked as follows: Denote the feasible +region of (A.9) as F1 and the feasible region of (A.10) as F2. From the above discussion, +it easily follows that F1 ⊆ F2. For the other direction, we perform integration by parts +for (A.10a), (A.10b), (A.10c), (A.10d) to obtain (A.9a), (A.9b), (A.9c), (A.9d) respectively. +Hence F2 ⊆ F1. It implies the equivalence of (A.9) and (A.10). +Finally, let p(x) = − f(x) +η ++ 1. Then (A.10) can be rewritten as +max +f +η +� ∞ +a +˜H(x)dp(x) +(A.11a) +s.t. +η +� ∞ +a +(x − a)dp(x) = β, +(A.11b) +η +� ∞ +a +˜G(x)dp(x) = Γ, +(A.11c) +η +� ∞ +a +dp(x) ≤ η, +(A.11d) +p(x) exists, non-decreasing and right-continuous for x ≥ a, +(A.11e) +0 ≤ p(x) ≤ 1 for x ≥ a, +(A.11f) +p(x) → 1 as x → ∞, +(A.11g) +p(x) = 0 for x < a, +(A.11h) +(1 − p(x))x → 0 for x → ∞, +(A.11i) +˜G(x)(1 − p(x)) → 0 as x → ∞. +(A.11j) +Since ˜H(x) = (x − a) = ˜G(x) = 0 at x = a, one can uniquely identify, up to mea- +sure zero, a non-decreasing, right-continuous p such that limx→∞ p(x) = 1 and p(x) = 0 +for x < a with a probability measure supported on [a, ∞). Finally, by Lemma A.4, con- +straints (A.11i) and (A.11j) can be derived from other constraints in this optimization +problem. Constraint (A.11d) is included in (A.11f). We have the equivalent problem as +14 + +follows +max +f +η +� ∞ +−∞ +˜H(x)dp(x) +s.t. +η +� ∞ +−∞ +(x − a)dp(x) = β, +η +� ∞ +−∞ +˜G(x)dp(x) = Γ, +p(x) exists, non-decreasing and right-continuous for x ∈ R, +0 ≤ p(x) ≤ 1 for x ∈ R, +p(x) → 1 as x → ∞, +p(x) = 0 for x < a. +This concludes the proof the first half of the theorem. +For the second half of the theorem, we first consider η = ¯η = η and rewrite the program +P +� +h, g, {Γ}, P2 +a,η,η,ν +� +as +max +f +� ∞ +a +h(x)f(x)dx +s.t. +� ∞ +a +f(x)dx = β, +� ∞ +a +g(x)f(x)dx = Γ, +f(a) = f(a+) = η, +f ′ ++(a) ≥ −ν, +f convex for x ≥ a, +f(x) ≥ 0 for x ≥ a. +(A.12) +Based on Lam and Mottet (2017), the formulation (A.12) is equivalent to +max +f +� ∞ +a +h(x)f(x)dx +(A.13a) +s.t. +� ∞ +a +f(x)dx = β, +(A.13b) +� ∞ +a +g(x)f(x)dx = Γ, +(A.13c) +15 + +f(a) = η, +(A.13d) +f ′ ++(x) exists and is non-decreasing and right-continuous for x ≥ a, +(A.13e) +− ν ≤ f ′ ++(x) ≤ 0 for x ≥ a, +(A.13f) +f ′ ++(x) → 0 a x → ∞, +(A.13g) +f(x) = +� x +a +f ′ ++(t)dt + η for x ≥ a. +(A.13h) +Here f(a+) denotes the right limit at a, and f(a) = f(a+) means that f is right-continuous +at a, implying a continuous extrapolation at a. +Denote +˜H(x) = +� x +a +h(u)du, +H(x) = +� x +a +˜H(u)du, +˜G(x) = +� x +a +g(u)du, +G(x) = +� x +a +˜G(u)du. +Consider the objective function (A.13a). Since ˜H and H are continuous, f is absolutely +continuous by (A.13h) and f ′ ++ has bounded variation because of (A.13f) and (A.13g), we +have, using integration by parts, +� ∞ +a +f(x)h(x)dx = +� ∞ +a +f(x)d ˜H(x) += f(x) ˜H(x)|∞ +a − +� ∞ +a +˜H(x)f ′ ++(x)dx += − +� ∞ +a +˜H(x)f ′ ++(x)dx += −H(x)f ′ ++(x)|∞ +a + +� ∞ +a +H(x)df ′ ++(x) += +� ∞ +a +H(x)df ′ ++(x) +(A.14) +where the third equality follows from Lemma A.2 with α = 0 and that ˜H(x) = O(x) as +x → ∞ since h is bounded. The fifth equality follows from Lemma A.2 again with α = 0 +and that H(x) = O(x2) as x → ∞. +For (A.13b), we can write +� ∞ +a +f(x)dx = +� ∞ +a +f(x)d(x − a) +16 + += f(x)(x − a)|∞ +a − +� ∞ +a +(x − a)f ′ ++(x)dx += − +� ∞ +a +(x − a)f ′ ++(x)dx += −(x − a)2 +2 +f ′ ++(x)|∞ +a + +� ∞ +a +(x − a)2 +2 +df ′ ++(x) += +� ∞ +a +(x − a)2 +2 +df ′ ++(x) +by merely viewing h ≡ 1 in (A.14). +For (A.13d), note that f(x) → 0 as x → ∞ from Lemma A.2 with α = 0, we can use +integration by parts again to write +f(a) = − +� ∞ +a +f ′ ++(x)dx = − +� ∞ +a +f ′ ++(x)d(x − a) += −f ′ ++(x)(x − a)|∞ +a + +� ∞ +a +(x − a)df ′ ++(x) += +� ∞ +a +(x − a)df ′ ++(x). +For (A.13c), since ˜G and G(x) are continuous, we have, using integration by parts, +� ∞ +a +f(x)g(x)dx = +� ∞ +a +f(x)d ˜G(x) += f(x) ˜G(x)|∞ +a − +� ∞ +a +˜G(x)f ′ ++(x)dx += − +� ∞ +a +˜G(x)f ′ ++(x)dx += −G(x)f ′ ++(x)|∞ +a + +� ∞ +a +G(x)df ′ ++(x) += +� ∞ +a +G(x)df ′ ++(x) +where the third equality and fifth equality follow from Lemma A.3. Therefore, (A.13) is +equivalent to +max +f +� ∞ +a +H(x)df ′ ++(x) +(A.15a) +s.t. +� ∞ +a +(x − a)2 +2 +df ′ ++(x) = β, +(A.15b) +� ∞ +a +(x − a)df ′ ++(x) = η, +(A.15c) +17 + +� ∞ +a +G(x)df ′ ++(x) = Γ, +(A.15d) +f ′ ++(x) exists and is non-decreasing and right-continuous for x ≥ a, +(A.15e) +− ν ≤ f ′ ++(x) ≤ 0 for x ≥ a, +(A.15f) +f ′ ++(x) → 0 as x → ∞, +(A.15g) +f(x) = +� x +a +f ′ ++(t)dt + η for x ≥ a, +(A.15h) +xf(x), x2f ′ ++(x) → 0 as x → ∞, +(A.15i) +˜G(x)f(x), G(x)f ′ ++(x) → 0 as x → ∞ +(A.15j) +and the constraint (A.15h) states that f can be recovered from f(x) = +� x +a f ′ ++(t)dt+η. Note +that this definition of f has a right derivative coinciding with the obtained f ′ ++(x). +The equivalence of (A.13) and (A.15) can be checked as follows: Denote the feasible +region of (A.13) as F1 and the feasible region of (A.15) as F2. From the above discus- +sion, it easily follows that F1 ⊆ F2. For the other direction, we perform integration by +parts for (A.15a), (A.15b), (A.15c), (A.15d) to obtain (A.13a), (A.13b), (A.13d), (A.13c) +respectively. Hence F2 ⊆ F1. It implies the equivalence of (A.13) and (A.15). +We now show that (A.15i) and (A.15j) are redundant. For f ′ ++(x) satisfying (A.15e), +(A.15f) and (A.15g), we know from (A.15b) that +� ∞ +x +(t − a)2 +2 +df ′ ++(t) → 0 as x → ∞. +Note that with the non-decreasing property of f ′ ++(x) via (A.15e), we have the following +inequality +� ∞ +x +(t − a)2 +2 +df ′ ++(t) ≥ (x − a)2 +2 +� ∞ +x +df ′ ++(t) ≥ 0. +Hence, with f ′ ++(x) → 0 as x → ∞ via (A.15g), we have +(x − a)2 +2 +f ′ ++(x) → 0 as x → ∞. +(A.16) +18 + +For (A.15c), we can write +η = +� ∞ +a +(x − a)df ′ ++(x) = (x − a)f ′ ++(x)|∞ +a − +� ∞ +a +f ′ ++(x)dx = +� ∞ +a +−f ′ ++(x)dx +where the third equality follows from (A.16). It is easy to conclude that +� ∞ +x +−f ′ ++(t)dt → 0 as x → ∞. +(A.17) +From (A.15h) we can easily derive from (A.15f) and (A.17) that +f(x) = +� x +a +f ′ ++(t)dt + η = +� x +a +f ′ ++(t)dt + +� ∞ +a +−f ′ ++(x)dx = +� ∞ +x +−f ′ ++(t)dt ≥ 0 (A.18) +=⇒ f(x) → 0 as x → ∞. +Then from (A.15b), we can write +β = +� ∞ +a +(x − a)2 +2 +df ′ ++(x) += f ′ ++(x)(x − a)2 +2 +|∞ +a − +� ∞ +a +f ′ ++(x)(x − a)dx += +� ∞ +a +−f ′ ++(x)(x − a)dx +where the third equality follows from (A.16). Then it is easy to conclude that +� ∞ +x +−f ′ ++(t)(t − a)dt → 0 as x → ∞. +Note that with the non-positive property of f ′ ++(x) via (A.15f), we have the following in- +equality +� ∞ +x +−f ′ ++(t)(t − a)dt ≥ (x − a) +� ∞ +x +−f ′ ++(t)dt ≥ 0. +Hence, with equation of f(x) in (A.18), we have +(x − a)f(x) → 0 as x → ∞ +which concludes the redundancy of constraint (A.15i). +Now we show the redundancy of constraint (A.15j). We first consider g(x) is a non- +negative function. Then G(x) and ˜G(x) are non-decreasing non-negative continuous func- +tions. +From (A.15d), we have +� ∞ +x +G(t)df ′ ++(t) → 0 as x → ∞. +19 + +Note that with the non-decreasing property of f ′ ++(x) via (A.15e) and G(x), we have the +following inequality +� ∞ +x +G(t)df ′ ++(t) ≥ G(x) +� ∞ +x +df ′ ++(t) ≥ 0. +Hence, with f ′ ++(x) → 0 as x → ∞ via (A.15g), we have +G(x)f ′ ++(x) → 0 as x → ∞. +(A.19) +For (A.15d), we can write +Γ = +� ∞ +a +G(x)df ′ ++(x) += f ′ ++(x)G(x)|∞ +a − +� ∞ +a +f ′ ++(x) ˜G(x)dx += +� ∞ +a +−f ′ ++(x) ˜G(x)dx +where the third equality follows from (A.19). Then it leads to +� ∞ +x +−f ′ ++(t) ˜G(t)dt → 0 as x → ∞. +Note that with the non-positive property of f ′ ++(x) via (A.15f) and the non-decreasing +property of ˜G(x), we have the following inequality +� ∞ +x +−f ′ ++(t) ˜G(t)dt ≥ ˜G(x) +� ∞ +x +−f ′ ++(t)dt ≥ 0. +Hence, with equation of f(x) in (A.18), we have +˜G(x)f(x) → 0 as x → ∞. +Now we consider the case when g(x) is a bounded-below function for x ≥ a and the +value of g(x) can be negative. We consider ˜g(x) = g(x) + | minx≥a g(x)|. Clearly, ˜g(x) and +| minx≥a g(x)| are non-negative functions, which implies we can use the results above. The +results for g(x) hence follow by linearity of integration and sum law of limits. A detailed +exposition is as follows: +Given +˜G(x) = +� x +a +˜g(t)dt + +� x +a +−| min +x≥a g(x)|dt += +� x +a +˜g(t)dt − (x − a)| min +x≥a g(x)|, +20 + +G(x) = +� x +a +� v +a +˜g(u)dudv − (x − a)2 +2 +| min +x≥a g(x)|, +we have +lim +x→∞ +� x +a +˜g(t)dtf(x) → 0, lim +x→∞ −(x − a)| min +x≥a g(x)|f(x) → 0 +=⇒ +lim +x→∞ +˜G(x)f(x) → 0, +lim +x→∞ +� x +a +� v +a +˜g(u)dudvf ′ ++(x) → 0, lim +x→∞ −(x − a)2 +2 +| min +x≥a g(x)|f ′ ++(x) → 0 +=⇒ +lim +x→∞ G(x)f(x) → 0, +which concludes the redundancy of constraint (A.15j). +Finally, let p(x) = +f′ ++(x) +ν ++ 1. Then (A.15) is equivalent to +max +p +� ∞ +a +νH(x)dp(x) +s.t. +� ∞ +a +ν (x − a)2 +2 +dp(x) = β, +� ∞ +a +ν(x − a)dp(x) = η, +p(x) exists and is non-decreasing and right-continuous for x ≥ a, +0 ≤ p(x) ≤ 1 for x ≥ a, +p(x) → 1 for x → ∞, +� ∞ +a +νG(x)dp(x) = Γ +(A.20) +21 + +or equivalently +max +p +� ∞ +−∞ +νH(x)dp(x) +s.t. +� ∞ +−∞ +ν (x − a)2 +2 +dp(x) = β, +� ∞ +−∞ +ν(x − a)dp(x) = η, +p(x) exists and is non-decreasing and right-continuous for x ∈ R, +0 ≤ p(x) ≤ 1 for x ∈ R, +p(x) → 1 for x → ∞, +p(x) = 0 for x < a, +� ∞ +−∞ +νG(x)dp(x) = Γ. +(A.21) +Since H(x) = (x − a)2 = (x − a) = G(x) = 0 at x = a, one can uniquely identify, up to +measure zero, a non-decreasing, right-continuous p such that limx→∞ p(x) = 1 and p(x) = 0 +for x < a with a probability measure supported on [a, ∞). Hence (A.21) is equivalent to +(A.20). +When η ̸= ¯η, one can replace the equality constraint +� ∞ +−∞ ν(x − a)dp(x) = η in (A.21) +by η ≤ +� ∞ +−∞ ν(x − a)dp(x) ≤ ¯η which forms a rectangular constraint. Note that the above +derivation holds true for any choice of β and Γ so that the result still holds when replacing +{Γ} = {(β, Γ)⊤} with general S. This concludes the result. +C.14 +Proof of Theorem 4.1: Choquet Method +Proof. Without loss of generality, we assume that g = +� +I(x ≥ a), g(x) +�⊤ and S = {Γ} = +{(β, Γ)⊤} for some scalars β, Γ where g(x) : [a, ∞) → R is an integrable function over +[a, ∞) and then we show how the result can be generalized to any region S. Notice that +22 + +for any feasible solution f in program (A.9), we can utilize Theorem A.1 and obtain +� ∞ +a +h(x)f(x)dx = β +� ∞ +a +� ∞ +0+ +h(x)I(a ≤ x < a + z) +z +dQ(z)dx += β +� ∞ +0+ +� ∞ +a +h(x)I(a ≤ x < a + z) +z +dxdQ(z) += β +� ∞ +0+ +� a+z +a +h(x)dx +z +dQ(z) = β +� ∞ +0+ +˜H(z + a) +z +dQ(z), +� ∞ +a +g(x)f(x)dx = β +� ∞ +0+ +˜G(z + a) +z +dQ(z), +� ∞ +a +f(x)dx = β +� ∞ +a +� ∞ +0+ +I(a ≤ x < a + z) +z +dQ(z)dx += β +� ∞ +0+ +� ∞ +a +I(a ≤ x < a + z) +z +dxdQ(z) += β +� ∞ +0+ +dQ(z) = β, +f(a) = +� ∞ +0+ +I(a ≤ x < a + z) +z +dQ(z) = +� ∞ +0+ +1 +zdQ(z). +Therefore, program (A.9) is equivalent to the following program, +max +Z∼Q βEQ +� ˜H(Z + a) +Z +� +s.t. +βEQ[1] = β, +EQ +� ˜G(Z + a) +Z +� += Γ +β , +EQ +� 1 +Z +� +≤ η +β , +Q ∈ M +(0, ∞) +where M +(0, ∞) is the space of non-negative bounded measures on (0, ∞). +Since EQ[ 1 +Z] < ∞, we can define a distribution function ˜Q ∈ M +(0, ∞) absolutely +continuous with respect to Q via d ˜Q +dQ(z) = β +η +1 +z. We convert the decision variable from Q +to ˜Q. Furthermore, the feasible set can be further restricted to P[0, ∞). That is because +the functions ˜H(z + a), ˜G(z + a), z are by construction equal to 0 at z = 0. Hence we +can always add an arbitrary mass at 0 to reach the upper bound of E ˜Q[1]. This in turn +deduces that upon proper normalization of the measure we can impose the constraint that +23 + +E ˜Q[1] = 1. Finally, we let x = z +a and obtain that the the following program is equivalent +to program (A.9): +max +X∼ ˜Q ηE ˜Q[ ˜H(X)] +s.t. +ηE ˜Q[X − a] = β, +ηE ˜Q[ ˜G(X)] = Γ, +˜Q ∈ P[a, ∞). +For the second part of the theorem, we first consider η = ¯η = η. For any feasible solution +f in program (A.12), Lemma 1 of Lam and Mottet (2017) implies that f is non-increasing. +Then based on Lemma 5.1 of Popescu (2005), f(x), ∀x ≥ a can be written as a generalized +mixture of right a-triangular density, i.e., +f(x) = β +� ∞ +0 +(a + z − x)+ +z2/2 +dQ(z) +where Q is a probability measure on [0, ∞). Since f exists everywhere for x ≥ a, we have +Q(z = 0) = 0. +Then since ˜H(x), ˜G(x), (a + z − x) are all absolutely continuous, we have, using inte- +gration by parts, +� ∞ +a +h(x)f(x)dx = β +� ∞ +a +h(x) +� ∞ +0 +(a + z − x)+ +z2/2 +dQ(z)dx += β +� ∞ +0 +� ∞ +a h(x)(a + z − x)+dx +z2/2 +dQ(z) += β +� ∞ +0 +� a+z +a +h(x)(a + z − x)dx +z2/2 +dQ(z) += β +� ∞ +0 +˜H(x)(a + z − x)|x=a+z +x=a ++ +� a+z +a +˜H(x)dx +z2/2 +dQ(z) += β +� ∞ +0 +� a+z +a +˜H(x)dx +z2/2 +dQ(z) = β +� ∞ +0+ +H(a + z) +z2/2 +dQ(z), +� ∞ +a +g(x)f(x)dx = β +� ∞ +0+ +G(a + z) +z2/2 +dQ(z), +� ∞ +a +f(x)dx = β +� ∞ +a +� ∞ +0 +(a + z − x)+ +z2/2 +dQ(z)dx = β +� ∞ +0 +� ∞ +a (a + z − x)+dx +z2/2 +dQ(z) +24 + += β +� ∞ +0 +� a+z +a +x(a + z − x)dx +z2/2 +dQ(z) += β +� ∞ +0 +(x − a)(a + z − x)|x=a+z +x=a ++ +� a+z +a +(x − a)dx +z2/2 +dQ(z) += β +� ∞ +0 +� a+z +a +(x − a)dx +z2/2 +dQ(z) = β +� ∞ +0+ +z2/2 +z2/2dQ(z) = β, +f(a) = β +� ∞ +0+ +z +z2/2dQ(z), +−f ′ ++(a) = lim +δn↓0 +−f(a + δn) + f(a) +δn += β lim +δn↓0 +� ∞ +0 +z − (z − δn)+ +δnz2/2 +dQ(z) += β lim +δn↓0 +� ∞ +0+ +z − (z − δn)+ +δnz2/2 +dQ(z) = β +� ∞ +0+ +1 +z2/2dQ(z). +Since 0 ≤ z−(z−δn)+ +δnz2/2 +≤ z−(z−δn+1)+ +δn+1z2/2 +, the exchange of limit and integration is followed by +monotone convergence theorem. Therefore, program (A.12) is equivalent to the following +program, +max +Z∼Q βEQ +�H(Z + a) +Z2/2 +� +s.t. +βEQ[1] = β, +βEQ +�G(Z + a) +Z2/2 +� += Γ, +βEQ +� Z +Z2/2 +� += η, +βEQ[ +1 +Z2/2] ≤ ν, +Q ∈ M +(0, ∞). +Since EQ[ +1 +Z2/2] < ∞, we can define a distribution function ˜Q ∈ M +(0, ∞) absolutely +continuous with respect to Q via d ˜Q +dQ(z) = β +ν +1 +z2/2. We convert the decision variable from Q +to ˜Q. Furthermore, the feasible set can be further restricted to P[0, ∞). That is because +the functions H(z + a), G(z + a), z, z2/2 are by construction equal to 0 at z = 0. Hence +we can always add an arbitrary mass at 0 to reach the upper bound of E ˜Q[1]. This in turn +deduces that upon proper normalization of the measure we can impose the constraint that +E ˜Q[1] = 1. Finally, we let x = z + a and obtain that the following program is equivalent to +25 + +program (A.12): +max +X∼ ˜Q νE ˜Q[H(X)] +s.t. +νE ˜Q +�(X − a)2 +2 +� += β, +νE ˜Q[G(X)] = Γ, +νE ˜Q[X − a] = η, +˜Q ∈ P[a, ∞). +When η ̸= ¯η, one can replace the equality constraint νE ˜Q[X−a] = η by η ≤ νE ˜Q[X−a] ≤ +¯η which gives a rectangular constraint. Note that the above derivation holds true for any +choice of β and Γ so that the result still holds when replacing {Γ} = {(β, Γ)⊤} with general +S. It concludes the proof. +C.15 +Proof of Theorem 4.2 +Proof. Without loss of generality, we consider r = 1. For positive r other than 1, one could +replace Σ by rΣ and the rest of the derivation is the same. We can consider S to be pure +hyper-ellipsoid or pure hyper-rectangle separately and the final result is a combination +of both. For the hyper-ellipsoid scenario, we rewrite Ma +� +H, G, SE(µ, Σ, 1) +� +, with given +constant values µ, Σ, ¯µ, µ, a as +max +P +EP[H(X)] +s.t. +� +�� +Σ−1/2� +EP[G(X)] − µ +� +1 +� +�� ∈ K, +P ∈ P[a, ∞) +(A.22) +where Σ−1/2 is the lower-triangular square-root matrix of Σ−1 obtained by Cholesky de- +composition, and K denotes the second order cone {(x1, . . . , xd, xd+1)⊤ ∈ Rd+1 : xd+1 ≥ +� +x2 +1 + · · · + x2 +d}. +26 + +The dual problem is derived as follows: The first step is to build a Lagrangian for this +program (A.22). Notice that with variables λ ≥ 0, κ, P ∈ M +[a, ∞), the following holds +EP[H(X)] + λ +� +1 − ∥Σ−1/2(EP[G(X)] − µ)∥2 +� ++ κ(1 − EP[1]) +=EP[H(X)] + λ +� +1 − max +∥u∥2≤1 u⊤Σ−1/2(EP[G(X)] − µ) +� ++ κ(1 − EP[1]) +=EP[H(X)] + λ − max +∥u∥2≤λ u⊤Σ−1/2(EP[G(X)] − µ) + κ(1 − EP[1]) += min +∥u∥2≤λ λ + κ + u⊤Σ−1/2µ + EP[H(X) − u⊤Σ−1/2G(X) − κ] +where ∥ · ∥2 denotes the Euclidean norm in Rp. +We can write a Lagrangian as follows: +L(P, λ, u, κ) = λ + κ + u⊤Σ−1/2µ + EP[H(X) − u⊤Σ−1/2G(X) − κ]. +One can check that Pa +� +H, G, SE(µ, Σ) +� +can be derived from +max +P∈M +[a,∞) +min +∥u∥2≤λ,κ L(P, λ, u, κ) +and the corresponding dual problem can be derived from +min +κ,∥u∥2≤λ +max +P∈M +[a,∞) L(P, λ, u, κ). +We then write the dual problem as follows: +min +κ,u,λ +λ + κ + u⊤Σ−1/2µ +s.t. +− H(x) + u⊤Σ−1/2G(x) + κ ≥ 0, x ∈ [a, ∞), +� +�� +u +λ +� +�� ∈ K. +Then for the hyper-rectangle scenario, we rewrite Ma +� +H, G, SR(µ, ¯µ) +� +as +max +X∼P +EP[H(X)] +s.t. +EP[G(X)] ≤ ¯µ, +− EP[G(X)] ≤ −µ, +P ∈ P[a, ∞). +27 + +Consider the following Lagrangian, with variables P, λ1, λ2, κ +L(P, λ1, λ2, κ) = EP[H(X)] + λ⊤ +1 (¯µ − EP[G(X)]) + λ⊤ +2 (−µ + EP[G(X)]) + κ(1 − EP[1]) += λ⊤ +1 ¯µ − λ⊤ +2 µ + κ + EP[−λ⊤ +1 G(X) + λ⊤ +2 G(X) − κ + H(X)]. +Therefore, we have the dual problem as follows: +min +λ1≥0,λ2≥0,κ +λ⊤ +1 ¯µ − λ⊤ +2 µ + κ +s.t. +− H(x) + (λ1 − λ2)⊤G(x) + κ ≥ 0, x ∈ [a, ∞). +It concludes the result. +C.16 +Proof of Corollary 4.1 +Proof. Using Theorem 4.1, we can easily get moment-constrained transformations as fol- +lows: +P(h, g, SE(µ, Σ, r), P1 +a,η) → Ma(η ˜H, η ˜G, SE(µ, Σ, r)), +P(h, g, SR(µ, ¯µ), P1 +a,η) → Ma(η ˜H, η ˜G, SR(µ, ¯µ)), +P(h, g, SE(µ, Σ, r), P2 +a,η,¯η,ν) → Ma(νH, ν(x − a, G⊤)⊤, SR(η, ¯η) × SE(µ, Σ, r)), +P(h, g, SR(µ, ¯µ), P2 +a,η,¯η,ν) → Ma(νH, ν(x − a, G⊤)⊤, SR(η, ¯η) × SR(µ, ¯µ)). +Then using Theorem 4.2, we obtain the results in the corollaries. +C.17 +Proof of Theorem 4.3 +Proof. Applying Proposition 3.1 from Bertsimas and Popescu (2005), the constraints (30) +are equivalent to that there exists two positive semi-definite matrices V = [vij]i,j=0,··· ,k and +28 + +W = [wij]i,j=0,··· ,k such that +V , W ⪰ 0, +� +i,j:i+j=2l−1 +vij = 0, +l = 1, · · · , k, +� +i,j:i+j=2l +vij = +k +� +r=l +�r +l +� +y1rar−l, +l = 0, · · · , k, +� +i,j:i+j=2l−1 +wij = 0, +l = 1, · · · , k, +� +i,j:i+j=2l +wij = +l +� +m=0 +k+m−l +� +r=m +� r +m +��k − r +l − m +� +y2rbr−mcm, +l = 0, · · · , k. +(A.23) +Replace (29) with (A.23) and obtain the result. +C.18 +Proof of Theorem A.1 +Proof. The proof is inspired from Theorem 2 of Goberna et al. (2007). For the rest of +the poof, for any set S, we denote int(S) the interior of S, |S| the cardinality, cone(S) := +{ +m +� +i=1 +λixi : xi ∈ S, λi ∈ R+, m ∈ N} the conical hull. The set of all natural number is N. +F∗(P) denotes the optimal region for program P. We first review some definitions of semi- +infinite linear programming and relevant results that are used later. Further information +are found in Goberna et al. (1981); Goberna and L´opez (2000). +According to Theorem 3.1 +of Goberna et al. (1981), Dµ is FM if and only if the characteristic cone Kc(Dµ) is closed, +Kc(Dµ) := cone +�� +G0(x), G1(x), . . . , Gn(x), H(x) +�⊤, x ∈ Ω; (0n+1, −1) +� +. +The linear constraint system {a(x)⊤y ≥ b(x), x ∈ S} is canonically closed if (1) +∃y0, a(x)⊤y0 > b(x), ∀x ∈ S; (2) ∃α(x) : Ω → R++, the set {α(x) · (a(x)⊤, b(x))⊤, x ∈ Ω} +is compact. Dµ is FM if the constraint system is canonically closed (Corollay 3.1.1 of +Goberna et al. (1981)). It is because canonically closed implies that +cone +�� +G0(x), G1(x), . . . , Gn(x), H(x) +�⊤, x ∈ Ω +� +is closed, which is a sufficient condition for the closedness of Kc(Dµ). Then we have +29 + +Lemma A.1. Program P is FM if the constraint system is canonically closed. +Here we discuss the optimality and duality theory between Pµ and Dµ. Denote first +moment cone of Dµ as M(Dµ) := cone +�� +G0(x), G1(x), . . . , Gn(x) +�⊤, x ∈ Ω +� +. +Then for +F(Dµ) ̸= ∅, F(Dµ) is bounded if and only if M(Dµ) = Rn+1 (Theorem 9.1 of Goberna +and L´opez (2000)) and F∗(Dµ) is a nonempty bounded set if and only if µ ∈ int(M(Dµ)) +(Corollary 9.3.1 of Goberna and L´opez (2000)). According to Corolary 4.1.1 of Goberna +et al. (1981), given any problem P and the corresponding dual problem D, if the problem +D is a feasible FM, then the following statements are true: (1) v(D) = −∞ if and only if +F(P) = ∅; (2) v(D) > −∞ if and only if v(D) = v(P). +We denote Z∗ = v(Pµ). Assume dr ̸= 0 and ρ > 0 as dr = 0 or ρ = 0 are trivial +cases. Since µ ∈ int +� +M(Dµ) +� +, F∗(Dµ) is bounded. Combined with the fact that Dµ is an +FM system, we obtain Pµ and Dµ are solvable and strong duality holds (see the second +remark after Theorem 2 of Goberna et al. (2007)). +In addition, +v(Pµ+ρdr) ≤ min{ +� +j∈{0}∪[d] +yj(µj + ρdrj)|y ∈ F(Dµ)} ≤ min{ +� +j∈{0}∪[d] +yj(µj + ρdrj)|y ∈ F∗(Dµ)} += v(Dµ) + ρ min{dr⊤y|y ∈ F∗(Dµ)} = v(Pµ) + ρ min{dr⊤y|y ∈ F∗(Dµ)} +(A.24) +where the first inequality is based on weak duality and the last equality follows from +the strong duality between Pµ and Dµ. Let us consider an auxiliary problem P′ and its +corresponding dual problem D′ as follows. +30 + +(P′) : max P, π EP [H(X)] + Z∗π +s.t. +EP [Gj(X)] + µjπ = drj, ∀j ∈ {0} ∪ [d], +P ∈ M +(Ω). +(D′) : min +y +dr⊤y +s.t. +� +{0}∪[d] +yjGj(x) ≥ H(x), +∀x ∈ Ω, +� +{0}∪[d] +yjµj = Z∗. +Since F∗(Dµ) is non-empty and bounded which is a feasible region for D′, we have +M(D′) = Rn+1, implying that every dr ∈ int +� +M(D′) +� +and thereby strong duality holds. +Moreover, P′ is solvable since the probability measure as the decision variables has compact +support. The above two arguments implies that dr⊤y can achieve its minimum value on +the feasible region F∗(Dµ). Let ( ¯P, ¯π) be an optimal solution of P′ and P ∗ be an optimal +solution of Pµ. +We argue that for any sufficiently small ρ > 0, P ∗ + ρ( ¯P + ¯πP ∗) is +feasible for Pµ+ρdr. If ¯π ≥ 0 , ρ can be any positive number. If ¯π < 0, ρ ≤ − 1 +¯π. To +see that, (1) P ∗ + ρ( ¯P + ¯πP ∗) ∈ M+(Ω) and (2). EP ∗+ρ( ¯P+¯πP ∗)[Gj(X)] = EP ∗[Gj(X)] + +Eρ( ¯P+¯πP ∗)[Gj(X)] = EP ∗[Gj(X)]+ρE ¯P[Gj(X)]+ρ¯πEP ∗[Gj(X)] = µj +ρ(drj −µj¯π+¯πµj) = +µj + ρdrj, j ∈ {0} ∪ [d]. +With strong duality of P′ and D′, we have +v(Pµ+ρdr) ≥ EP ∗+ρ( ¯P+¯πP ∗)[H(X)] += EP ∗[H(X)] + ρE ¯P[H(X)] + ρ¯πEP ∗[H(X)] += v(Pµ) + ρE ¯P[H(X)] + ρ¯πZ∗ = v(Pµ) + v(P′) += v(Pµ) + ρ min{dr⊤y|y ∈ F∗(Dµ)}. +(A.25) +Combining (A.24) and (A.25) we have +v(Pµ) + ρ min{dr⊤y|y ∈ F∗(Dµ)} ≥ v(Pµ+ρdr) ≥ v(Pµ) + ρ min{dr⊤y|y ∈ F∗(Dµ).} +We then have +v(Pµ+ρdr) = v(Pµ) + ρ min{dr⊤y|y ∈ F∗(Dµ), } +31 + +which concludes the proof. +C.19 +Some Useful Theorems and Lemmas +Lemma A.2. Given α ∈ R, if (i) +� ∞ +a xαf(x)dx = Γ, (ii) f(x) is non-increasing for x ≥ a, +(iii) f(x) ≥ 0, ∀x ≥ a, then we have xα+1f(x) → 0 as x → ∞. +Besides conditions above, if we have f ′ ++(x) is non-decreasing, non-positive for x ≥ a +and f(x) = +� x +a f ′ ++(t)dt + η, then we have xα+2f ′ ++(x) → 0 as x → ∞. +Proof of Lemma A.2. For α ≤ 0, xα becomes bounded function, the results easily follow. +Now we assume α > 0. Consider the function +T(x) = xα+1f(x) − (α + 1) +� x +a +tαf(t)dt. +For any (a ∨ 0) ≤ x1 ≤ x2, +T(x2) − T(x1) = xα+1 +2 +f(x2) − xα+1 +1 +f(x1) − +� x2 +x1 +f(t)dtα+1 +≤ xα+1 +2 +f(x2) − xα+1 +1 +f(x1) − f(x2)(xα+1 +2 +− xα+1 +1 +) += −xα+1 +1 +f(x1) + xα+1 +1 +f(x2) += xα+1 +1 +(f(x2) − f(x1)) ≤ 0. +From above, we have T(x) is non-increasing for x ≥ a ∨ 0. Since xα+1f(x) ≥ 0 and +0 ≤ +� x +a tαf(t)dt ≤ Γ for x ≥ a ∨ 0, we have that T(x) is bounded from below and thereby +converges to a limit. Given +� x +a tαf(t)dt → Γ as x → ∞, we must have xα+1f(x) go to +some finite non-negative value. If xα+1f(x) ≥ ϵ > 0 for all large enough x. This means +xαf(x) ≥ +ϵ +x for all large enough x, and hence +� ∞ +a xαf(x)dx = ∞. It contradicts with +condition (ii) above. Therefore, xα+1f(x) must converge to 0. +To prove the second part, consider a function +P(x) = −xα+2f ′ ++(x) + (α + 2)[− +� ∞ +x +tα+1f ′ ++(t)dt]. +32 + +Note that because f is absolutely continuous and limx→∞ xα+1f(x) → 0 as we have just +proved, integration by parts yields that +− +� ∞ +x +tα+1f ′ ++(t)dt = xα+1f(x) + (α + 1) +� ∞ +x +tαf(t)dt < ∞, ∀x ≥ (a ∨ 0). +For any (a ∨ 0) ≤ x1 ≤ x2, +P(x2) − P(x1) = xα+2 +1 +f ′ ++(x1) − xα+2 +2 +f ′ ++(x2) + +� x2 +x1 +f ′ ++(t)dtα+2 +≤ xα+2 +1 +f ′ ++(x1) − xα+2 +2 +f ′ ++(x2) + f ′ ++(x2)(xα+2 +2 +− xα+2 +1 +) += xα+2 +1 +(f ′ ++(x1) − f ′ ++(x2)) ≤ 0 +From above, we have P(x) is non-increasing for x ≥ (a ∨ 0). Since −xα+2f ′ ++(x) ≥ 0 for +x ≥ (a ∨ 0) and − +� ∞ +x tαf ′ ++(t)dt is non-negative, we have that P(x) is bounded from below +and converges to a limit. Because limx→∞ xα+1f(x) → 0 and limx→∞ +� ∞ +x f(t)tαdt = 0, we +have limx→∞(− +� ∞ +x xα+1f ′ ++(t)dt) = 0. It implies −xα+2f ′ ++(x) must converge to some finite +non-negative value as x → ∞. +If −xα+2f ′ ++(x) ≥ ϵ > 0 for all large enough x. This means −xα+1f ′ ++(x) ≥ ϵ +x for all large +enough x, and hence − +� ∞ +x tα+1f ′ ++(t)dt = ∞ for x ≥ a. It contradicts with the finiteness of +the integration of −tα+1f ′ ++(t) above. Therefore, −xα+1f ′ ++(x) must converge to 0. +Lemma A.3. Given function g(x) bounded below for x ≥ a, if (i) +� ∞ +a g(x)f(x)dx = Γ, +(ii) f(x) is non-increasing for x ≥ a, (iii) f(x) ≥ 0, ∀x ≥ a is a density function, then we +have ˜G(x)f(x) → 0 as x → ∞,where ˜G(x) = +� x +a g(u)du. +Besides conditions above, if we have f ′ ++(x) is non-decreasing, non-positive for x ≥ a, and +f(x) = +� x +a f ′ ++(t)dt + η, then we have G(x)f ′ ++(x) → 0 as x → ∞, where G(x) = +� x +a ˜G(u)du. +Proof of Lemma A.3. We first assume function g(x) is a non-negative function for x ≥ a. +Consider the function +T(x) = ˜G(x)f(x) − +� x +a +g(t)f(t)dt. +33 + +For any a ≤ x1 ≤ x2, +T(x2) − T(x1) = ˜G(x2)f(x2) − ˜G(x1)f(x1) − +� x2 +x1 +g(t)f(t)dt += ˜G(x2)f(x2) − ˜G(x1)f(x1) − +� x2 +x1 +f(t)d ˜G(t) +≤ ˜G(x2)f(x2) − ˜G(x1)f(x1) − f(x2)( ˜G(x2) − ˜G(x1)) += − ˜G(x1)f(x1) + f(x2) ˜G(x1) += ˜G(x1)(f(x2) − f(x1)) ≤ 0. +From above, we have T(x) is non-increasing for x ≥ a. +Since ˜G(x)f(x) ≥ 0 and +0 ≤ +� x +a g(t)f(t)dt ≤ Γ for x ≥ a, we have T(x) bounded from below and converge to a +limit. Because +� x +a g(t)f(t)dt → Γ as x → ∞, ˜G(x)f(x) must go to some finite non-negative +value as x → ∞. We now show that ˜G(x)f(x) → 0 as x → ∞. +First of all, since ˜G(x) is non-negative and non-decreasing function, we must have that +˜G(x) either blows up or converges to some positive finite value. We first consider the case +when ˜G(x) converges to some positive finite value as x → ∞. Since f(x) goes to zeros as +x → ∞ (indeed, f(x) ≥ 0, f(x) is non-increasing for x ≥ a and f(x) is a density function), +we have ˜G(x)f(x) → 0 as x → ∞. +Now we consider the second case when ˜G(x) blows up as x → ∞. We prove by contra- +diction. Since ˜G(x)f(x) is non-negative for x ≥ a, we suppose that ˜G(x)f(x) will converge +to a positive value as x → ∞. It implies that there exist ϵ > 0 and xϵ > a such that +˜G(x)f(x) ≥ ϵ, ∀x ≥ xϵ +Hence we have +� ∞ +xϵ +f(x)g(x)dx ≥ +� ∞ +xϵ +ϵ +˜G(x) +g(x)dx += +� ∞ +xϵ +ϵ +˜G(x) +d ˜G(x) += +� ∞ +xϵ +ϵ d ln ˜G(x) +34 + += ϵ ln ˜G(x)|∞ − ϵ ln ˜G(xϵ) = ∞ +However, we have +� ∞ +xϵ +f(x)g(x)dx ≤ +� ∞ +a +f(x)g(x)dx = Γ < ∞. +We get a contradiction. Hence we must have ˜G(x)f(x) → 0 as x → ∞. +To prove the second part, we consider the function +P(x) = −G(x)f ′ ++(x) + [− +� ∞ +x +˜G(t)f ′ ++(t)dt]. +Note that because f is absolutely continuous and limx→∞ ˜G(x)f(x) → 0 as we have just +proved, integration by parts yields that +− +� ∞ +x +˜G(t)f ′ ++(t)dt = ˜G(x)f(x) + +� ∞ +x +f(t)g(t)dt < ∞. +For any a ≤ x1 ≤ x2, we have +P(x2) − P(x1) = −G(x2)f ′ ++(x2) + G(x1)f ′ ++(x1) + [ +� x2 +x1 +˜G(t)f ′ ++(t)dt] += −G(x2)f ′ ++(x2) + G(x1)f ′ ++(x1) + f ′ ++(x2)(G(x2) − G(x1)) += G(x1)(f ′ ++(x1) − f ′ ++(x2)) ≤ 0. +From above, we have P(x) is non-increasing for x ≥ a. Since −G(x)f ′ ++(x) ≥ 0 and +− +� ∞ +x +˜G(t)f ′ ++(t)dt ≥ 0 for x ≥ a, we have that P(x) is bounded from below and converges +to a limit. Because limx→∞ ˜G(x)f(x) and limx→∞ +� ∞ +x f(t)g(t)dt are both equal to zero, we +have limx→∞(− +� ∞ +x +˜G(t)f ′ ++(t)dt) = 0. It implies that −G(x)f ′ ++(x) must go to some finite +non-negative value as x → ∞. We now show that −G(x)f ′ ++(x) → 0 as x → ∞. +First of all, since G(x) is non-negative and non-decreasing function, we must have that +G(x) either blows up or converges to some positive finite value. We first consider the case +when G(x) converges to some positive finite value as x → ∞. Suppose that f ′ ++(x) ̸→ 0. +Then by non-positiveness and non-decreasing properties of f ′ ++(x), we have f ′ ++(x) → c < 0 +as x → ∞. +However, f(x) = +� x +a f ′ ++(t)dt + η → −∞ as x → ∞, violating the non- +negative condition of f(x). We then have f ′ ++(x) go to zeros as x → ∞, thereby implying +35 + +−G(x)f ′ ++(x) → 0 as x → ∞. +Now we consider the second case when G(x) blows up as x → ∞. We prove by con- +tradiction, since −G(x)f ′ ++(x) is non-negative for x ≥ 0, we suppose that −G(x)f ′ ++(x) will +converge to a positive value as x → ∞. It implies that there exist ϵ > 0 and xϵ > a such +that −G(x)f ′ ++(x) ≥ ϵ, ∀x ≥ xϵ +Hence we have +� ∞ +xϵ +−f ′ ++(x) ˜G(x)dx ≥ +� ∞ +xϵ +ϵ +G(x) +˜G(x)dx += +� ∞ +xϵ +ϵ +G(x)dG(x) += +� ∞ +xϵ +ϵ d ln G(x) += ϵ ln G(x)|∞ − ϵ ln G(xϵ) = ∞. +However, we have +� ∞ +xϵ +−f ′ ++(x) ˜G(x)dx = ˜G(xϵ)f(xϵ) + +� ∞ +xϵ +f(t)g(t)dt < ∞. +We get a contradiction. Hence we must have G(x)f ′ ++(x) → 0 as x → ∞. +Now we consider the case when g(x) is a bounded-below function for x ≥ a and the +value of g(x) can be negative. We consider ˜g(x) = g(x) + | minx≥a g(x)|. Clearly, ˜g(x) and +| minx≥a g(x)| is non-negative function, which implies we can use the results just above. +The results for g(x) hence follow by linearity of integration and sum law of limits. +Lemma A.4. If E[G(X)] = β where β is some finite constant and G(x) : R → R is a +non-decreasing function and bounded from below for x ≥ a with some constant a. For any +P ∈ P[a, ∞), we have EP[I(X ≥ x)]G(x) → 0 as x → ∞. +Proof of Lemma A.4. We first assume that G(x) is a non-negative function. +Based on +E[G(X)] = β, it is easy to obtain that E[G(X)I(X ≥ x)] → 0 as x → ∞. Then E[I(X ≥ +x)]G(x) ≤ E[G(X)I(X ≥ x)] → 0 as x → ∞. +The first inequality follows from the +36 + +non-decreasing property of G(x). And from the non-negative property of G(x), we have +E[I(X ≥ x)]G(x) ≥ 0. It implies that E[I(X ≥ x)]G(x) → 0 as x → ∞. +Now we consider the case when G(x) is a bounded-below function and the value of G(x) +can be negative. We consider ˜G(x) = G(x) + | minx≥a G(x)|. Here, the minimum value is +taken with respect to the support of the probability space. Clearly, ˜G(x) and | minx≥a G(x)| +are non-negative functions, which implies we can use the results above. The result for G(x) +hence follows by sum law of limits. Given limx→∞ E[I(X ≥ x)][G(x) + | minx≥a G(x)|] = 0 +and limx→∞ E[I(X ≥ x)][−| minx≥a G(x)|] = 0, we have limx→∞ E[I(X ≥ x)]G(x) = 0. +Theorem A.1. The following two statements are equivalent: +(1). A function f is finite, non-increasing and right-continuous for x ≥ a and +� ∞ +a f(x)dx = +β. +(2). f(x), x ≥ a is a generalized mixture of the indicator functions I(a≤x ∞, +i.e., +f(x) = β +� ∞ +0+ +I(a ≤ x < a + z) +z +dQ(z) +where Q is a probability measure on (0, ∞). +Proof of Theorem A.1. (1) =⇒ (2): +The probability distribution function defined as F(x) := 1 − β + +� x +a f(x)dx, ∀x ≥ a +is absolute continuous for x ≥ a. We consider another probability distribution ˜F where +˜F(x) = F(x)+β−1 +β +, x ≥ a and F(x) = 0, x < a. Then ˜F is still absolutely continuous on +x ≥ a and density ˜f(x) is zero for x < a and f(x) +β +for x ≥ a. Then ˜F is unimodal at point +a. Moreover, ˜F(x) = +� x +a ˜f(s)ds. The right derivative of ˜F(x) exists for x ≥ a. We also +notice that ˜f(x + δ) ≤ +� x+δ +x +˜f(s)ds +δ +≤ ˜f(x). Since f is right continuous for x ≥ a, we have +˜f(x) = lim +δ↓0 +˜f(x + δ) ≤ lim +δ↓0 +� x+δ +x +˜f(s)ds +δ +≤ ˜f(x). Therefore, +˜F ′ ++(x) = lim +δ↓0 +˜F(x + δ) − ˜F(x) +δ += lim +δ↓0 +� x+δ +x +˜f(s)ds +δ += ˜f(x). +37 + +Based on Theorem 1.2 of Dharmadhikari and Joag-Dev (1988), ˜F is a generalized mix- +ture of the distribution functions Wa,z where Wa,z denote the uniform distribution function +on (a, a + z), z > 0. The reason that we do not consider z ≤ 0 is that ˜F(x) = 0 for x < a +and ˜F is absolutely continuous on x ≥ a. In particular, we have +˜F(x) = +� ∞ +0+ +Wa,z(x)dQ(z) +where Q is a probability measure on (0, ∞). +We then have +˜f(x) = ˜F ′ ++(x) = lim +δn↓0 +˜F(x + δn) − ˜F(x) +δn += lim +δn↓0 +� ∞ +0+ +Wa,z(x + δn) − Wa,z(x) +δn +dQ(z) +(A.26) += lim +δn↓0 +� (x−a)− +0+ +Wa,z(x + δn) − Wa,z(x) +δn +dQ(z) ++ lim +δn↓0 +Wa,z(x + δn) − Wa,z(x) +δn +Q(x − a) + lim +δn↓0 +� ∞ +(x−a)+ +Wa,z(x + δn) − Wa,z(x) +δn +dQ(z) +(A.27) += lim +δn↓0 +Wa,x−a(x + δn) − Wa,x−a(x) +δn +Q(z = x − a) + +� ∞ +(x−a)+ +1 +zdQ(z), +(A.28) +and lim +δn↓0 +Wa,x−a(x+δn)−Wa,x−a(x) +δn += 0. Hence we have ˜f(x) = +� ∞ +(x−a)+ +1 +zdQ(z). +The exchange of limit and summation follows from observing that the limit in the +equality (A.26) and the first two terms of (A.27) exist. Since 0 ≤ +Wa,z(x+δn)−Wa,z(x) +δn +≤ +Wa,z(x+δn+1)−Wa,z(x) +δn+1 +, the exchange of limit and integration in the equality (A.28) follows +from monotone convergence theorem. +It then concludes with +� ∞ +0+ +I(a ≤ x < a + z) +z +dQ(z) = +� ∞ +(x−a)+ +1 +zdQ(z) = ˜f(x) +=⇒ +f(x) = β +� ∞ +0+ +I(a ≤ x < a + z) +z +dQ(z) +(2) =⇒ (1) : +Since +� ∞ +a +f(x)dx = +� ∞ +a +β +� ∞ +0+ +I(a ≤ x < a + z) +z +dQ(z)dx = β +� ∞ +0+ +� ∞ +a +I(a ≤ x < a + z) +z +dxdQ(z) +38 + += β +� ∞ +0+ +1dxdQ(z) = β. +For any x2 ≥ x1 ≥ a, we have ∀z > 0, I(a ≤ x1 < a + z) ≥ I(a ≤ x2 < a + z). It is easy +to obtain that +f(x1) = β +� ∞ +0+ +I(a ≤ x1 < a + z) +z +dQ(z) ≥ β +� ∞ +0+ +I(a ≤ x2 < a + z) +z +dQ(z) = f(x2). +Finally, f(x) is right continuous for x ≥ a because ∀x ≥ a, +lim +δ↓0 f(x + δ) = lim +δ↓0 β +� ∞ +0+ +I(a ≤ x + δ < a + z) +z +dQ(z) += β +� ∞ +0+ +lim +δ↓0 +I(a ≤ x + δ < a + z) +z +dQ(z) += β +� ∞ +0+ +I(a ≤ x < a + z) +z +dQ(z) = f(x). +Since I(a ≤ x < a + z) is a right continuous function and +�� I(a≤x